Основні числові системи/Цілі числа

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available" lang="uk" dir="ltr"> <head> <meta charset="UTF-8"> <title>Основні числові системи/Цілі числа — Вікіпідручник</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-disabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )ukwikibooksmwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":[",\t."," \t,"],"wgDigitTransformTable":["",""], "wgDefaultDateFormat":"dmy","wgMonthNames":["","січень","лютий","березень","квітень","травень","червень","липень","серпень","вересень","жовтень","листопад","грудень"],"wgRequestId":"46332db9-e106-427b-a217-67ae12e3a893","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Основні_числові_системи/Цілі_числа","wgTitle":"Основні числові системи/Цілі числа","wgCurRevisionId":36331,"wgRevisionId":36331,"wgArticleId":7719,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Основні числові системи"],"wgPageViewLanguage":"uk","wgPageContentLanguage":"uk","wgPageContentModel":"wikitext","wgRelevantPageName":"Основні_числові_системи/Цілі_числа","wgRelevantArticleId":7719,"wgIsProbablyEditable":true, "wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikibooks","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgVisualEditor":{"pageLanguageCode":"uk","pageLanguageDir":"ltr","pageVariantFallbacks":"uk"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":50000,"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q12503","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"]};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready", "user.options":"loading","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","wikibase.client.vector-2022","ext.checkUser.clientHints","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=uk&modules=ext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=uk&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=uk&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Основні числові системи/Цілі числа — Вікіпідручник"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//uk.m.wikibooks.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"> <link rel="alternate" type="application/x-wiki" title="Редагувати" href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit"> <link rel="icon" href="/static/favicon/wikibooks.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Вікіпідручник (uk)"> <link rel="EditURI" type="application/rsd+xml" href="//uk.wikibooks.org/w/api.php?action=rsd"> <link rel="canonical" href="https://uk.wikibooks.org/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.uk"> <link rel="alternate" type="application/atom+xml" title="Вікіпідручник — Atom-стрічка" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Основні_числові_системи_Цілі_числа rootpage-Основні_числові_системи skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Перейти до вмісту</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Головне меню" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Головне меню </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Головне меню</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">сховати</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Навігація </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Перейти на головну сторінку [z]" accesskey="z">Головна сторінка</a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D1%96%D0%B4%D1%80%D1%83%D1%87%D0%BD%D0%B8%D0%BA:%D0%9F%D0%BE%D1%80%D1%82%D0%B0%D0%BB_%D1%81%D0%BF%D1%96%D0%BB%D1%8C%D0%BD%D0%BE%D1%82%D0%B8" title="Про проєкт, про те, що Ви можете зробити, і що де шукати">Портал спільноти</a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D1%96%D0%B4%D1%80%D1%83%D1%87%D0%BD%D0%B8%D0%BA:%D0%9F%D0%BE%D1%82%D0%BE%D1%87%D0%BD%D1%96_%D0%BF%D0%BE%D0%B4%D1%96%D1%97" title="Список поточних подій">Поточні події</a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9D%D0%BE%D0%B2%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F" title="Список останніх змін у цій вікі [r]" accesskey="r">Нові редагування</a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D0%B8%D0%BF%D0%B0%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" title="Переглянути випадкову сторінку [x]" accesskey="x">Випадкова стаття</a></li><li id="n-help" class="mw-list-item"><a href="https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Contents" title="Довідка з проєкту">Довідка</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/%D0%93%D0%BE%D0%BB%D0%BE%D0%B2%D0%BD%D0%B0_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B0" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikibooks.svg" alt="" aria-hidden="true" height="50" width="50"> <img class="mw-logo-wordmark" alt="Вікіпідручник" src="/static/images/mobile/copyright/wikibooks-wordmark-uk.svg" style="width: 7.5em; height: 0.875em;"> <img class="mw-logo-tagline" alt="" src="/static/images/mobile/copyright/wikibooks-tagline-uk.svg" width="120" height="9" style="width: 7.5em; height: 0.5625em;"> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%88%D1%83%D0%BA" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Шукати у Вікіпідручника [f]" accesskey="f"> Пошук </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Пошук у Вікіпідручнику" aria-label="Пошук у Вікіпідручнику" autocapitalize="sentences" title="Шукати у Вікіпідручника [f]" accesskey="f" id="searchInput" > </div> <input type="hidden" name="title" value="Спеціальна:Пошук"> </div> <button class="cdx-button cdx-search-input__end-button">Знайти</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Особисті інструменти"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Зовнішній вигляд"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Змінити зовнішній вигляд розміру, ширини та кольору сторінки" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Зовнішній вигляд" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Зовнішній вигляд </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_uk.wikibooks.org&uselang=uk" class="">Пожертвувати</a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D1%82%D0%B2%D0%BE%D1%80%D0%B8%D1%82%D0%B8_%D0%BE%D0%B1%D0%BB%D1%96%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Пропонуємо створити обліковий запис і увійти в систему; однак, це не обов'язково" class="">Створити обліковий запис</a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D1%85%D1%96%D0%B4&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Заохочуємо Вас увійти в систему, але це необов'язково. [o]" accesskey="o" class="">Увійти</a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Більше опцій" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Особисті інструменти" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Особисті інструменти </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="Меню користувача" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_uk.wikibooks.org&uselang=uk">Пожертвувати</a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D1%82%D0%B2%D0%BE%D1%80%D0%B8%D1%82%D0%B8_%D0%BE%D0%B1%D0%BB%D1%96%D0%BA%D0%BE%D0%B2%D0%B8%D0%B9_%D0%B7%D0%B0%D0%BF%D0%B8%D1%81&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Пропонуємо створити обліковий запис і увійти в систему; однак, це не обов'язково"> Створити обліковий запис</a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%92%D1%85%D1%96%D0%B4&returnto=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Заохочуємо Вас увійти в систему, але це необов'язково. [o]" accesskey="o"> Увійти</a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Сторінки для редакторів, які не ввійшли в систему <a href="/wiki/%D0%94%D0%BE%D0%B2%D1%96%D0%B4%D0%BA%D0%B0:%D0%92%D1%81%D1%82%D1%83%D0%BF" aria-label="Дізнатися більше про редагування">дізнатися більше</a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D1%96%D0%B9_%D0%B2%D0%BD%D0%B5%D1%81%D0%BE%D0%BA" title="Список редагувань, зроблених з цієї IP-адреси [y]" accesskey="y">Внесок</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9C%D0%BE%D1%94_%D0%BE%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F" title="Обговорення редагувань з цієї IP-адреси [n]" accesskey="n">Обговорення</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Сайт"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Зміст" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Зміст</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">сховати</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Вступ</div> </a> </li> <li id="toc-Абсолютна_величина" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Абсолютна_величина"> <div class="vector-toc-text"> 1 Абсолютна величина </div> </a> <ul id="toc-Абсолютна_величина-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Цілі_вирази" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Цілі_вирази"> <div class="vector-toc-text"> 2 Цілі вирази </div> </a> <ul id="toc-Цілі_вирази-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Метод_порівняння" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Метод_порівняння"> <div class="vector-toc-text"> 3 Метод порівняння </div> </a> <ul id="toc-Метод_порівняння-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Тотожні_вирази" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Тотожні_вирази"> <div class="vector-toc-text"> 4 Тотожні вирази </div> </a> <ul id="toc-Тотожні_вирази-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Степінь_числа_з_натуральним_показником" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Степінь_числа_з_натуральним_показником"> <div class="vector-toc-text"> 5 Степінь числа з натуральним показником </div> </a> <ul id="toc-Степінь_числа_з_натуральним_показником-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Мономи_(одночлени)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Мономи_(одночлени)"> <div class="vector-toc-text"> 6 Мономи (одночлени) </div> </a> <ul id="toc-Мономи_(одночлени)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Поліноми_(многочлени)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Поліноми_(многочлени)"> <div class="vector-toc-text"> 7 Поліноми (многочлени) </div> </a> <ul id="toc-Поліноми_(многочлени)-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Зміст" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Сховати/показати зміст" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Сховати/показати зміст </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Основні числові системи/Цілі числа</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Перейдіть до статті іншою мовою. Доступний у 3 мовах" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-3" aria-hidden="true" > 3 мови </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikibooks.org/wiki/A_Guide_to_the_GRE/Integers" title="A Guide to the GRE/Integers — англійська" lang="en" hreflang="en" data-title="A Guide to the GRE/Integers" data-language-autonym="English" data-language-local-name="англійська" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikibooks.org/wiki/%E9%AB%98%E7%AD%89%E5%AD%A6%E6%A0%A1%E6%95%B0%E5%AD%A6A/%E6%95%B0%E5%AD%A6%E3%81%A8%E4%BA%BA%E9%96%93%E3%81%AE%E6%B4%BB%E5%8B%95" title="高等学校数学A/数学と人間の活動 — японська" lang="ja" hreflang="ja" data-title="高等学校数学A/数学と人間の活動" data-language-autonym="日本語" data-language-local-name="японська" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikibooks.org/wiki/%E6%95%B4%E6%95%B8" title="整數 — китайська" lang="zh" hreflang="zh" data-title="整數" data-language-autonym="中文" data-language-local-name="китайська" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12503#sitelinks-wikibooks" title="Редагувати міжмовні посилання" class="wbc-editpage">Редагувати посилання</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Простори назв"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Вміст статті [c]" accesskey="c">Сторінка</a></li><li id="ca-talk" class="new vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D0%B1%D0%B3%D0%BE%D0%B2%D0%BE%D1%80%D0%B5%D0%BD%D0%BD%D1%8F:%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&redlink=1" rel="discussion" class="new" title="Обговорення сторінки (такої сторінки не існує) [t]" accesskey="t">Обговорення</a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Змінити варіант мови" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >українська </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Перегляди"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0">Читати</a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e">Редагувати</a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history" title="Журнал змін сторінки [h]" accesskey="h">Переглянути історію</a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Інструменти сторінки"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Інструменти" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" >Інструменти </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Інструменти</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">сховати</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="Більше опцій" > <div class="vector-menu-heading"> Дії </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0">Читати</a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit" title="Редагувати цю сторінку [e]" accesskey="e">Редагувати</a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=history">Переглянути історію</a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Загальний </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D1%81%D0%B8%D0%BB%D0%B0%D0%BD%D0%BD%D1%8F_%D1%81%D1%8E%D0%B4%D0%B8/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Перелік усіх сторінок, які посилаються на цю сторінку [j]" accesskey="j">Посилання сюди</a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9F%D0%BE%D0%B2%27%D1%8F%D0%B7%D0%B0%D0%BD%D1%96_%D1%80%D0%B5%D0%B4%D0%B0%D0%B3%D1%83%D0%B2%D0%B0%D0%BD%D0%BD%D1%8F/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" rel="nofollow" title="Останні зміни на сторінках, на які посилається ця сторінка [k]" accesskey="k">Пов'язані редагування</a></li><li id="t-upload" class="mw-list-item"><a href="//commons.wikimedia.org/wiki/Special:UploadWizard?uselang=uk" title="Завантажити файли [u]" accesskey="u">Завантажити файл</a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%81%D1%82%D0%BE%D1%80%D1%96%D0%BD%D0%BA%D0%B8" title="Перелік спеціальних сторінок [q]" accesskey="q">Спеціальні сторінки</a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&oldid=36331" title="Постійне посилання на цю версію цієї сторінки">Постійне посилання</a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=info" title="Додаткові відомості про цю сторінку">Інформація про сторінку</a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%A6%D0%B8%D1%82%D0%B0%D1%82%D0%B0&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&id=36331&wpFormIdentifier=titleform" title="Інформація про те, як цитувати цю сторінку">Цитувати сторінку</a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:UrlShortener&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%25A6%25D1%2596%25D0%25BB%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0">Отримати вкорочену URL-адресу</a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:QrCode&url=https%3A%2F%2Fuk.wikibooks.org%2Fwiki%2F%25D0%259E%25D1%2581%25D0%25BD%25D0%25BE%25D0%25B2%25D0%25BD%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25BE%25D0%25B2%25D1%2596_%25D1%2581%25D0%25B8%25D1%2581%25D1%2582%25D0%25B5%25D0%25BC%25D0%25B8%2F%25D0%25A6%25D1%2596%25D0%25BB%25D1%2596_%25D1%2587%25D0%25B8%25D1%2581%25D0%25BB%25D0%25B0">Завантажити QR-код</a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Друк/експорт </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%BD%D0%B8%D0%B3%D0%B0&bookcmd=book_creator&referer=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96+%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96+%D1%87%D0%B8%D1%81%D0%BB%D0%B0">Створити книгу</a></li><li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:DownloadAsPdf&page=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8%2F%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=show-download-screen">Завантажити як PDF</a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&printable=yes" title="Версія цієї сторінки для друку [p]" accesskey="p">Версія до друку</a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> В інших проєктах </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Integers" hreflang="en">Вікісховище</a></li><li class="wb-otherproject-link wb-otherproject-wikifunctions mw-list-item"><a href="https://www.wikifunctions.org/wiki/Z16683" hreflang="en">Вікіфункції</a></li><li class="wb-otherproject-link wb-otherproject-wikipedia mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A6%D1%96%D0%BB%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" hreflang="uk">Вікіпедія</a></li><li class="wb-otherproject-link wb-otherproject-wikiquote mw-list-item"><a href="https://uk.wikiquote.org/wiki/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" hreflang="uk">Вікіцитати</a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q12503" title="Посилання на пов’язаний елемент сховища даних [g]" accesskey="g">Елемент Вікіданих</a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Інструменти сторінки"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Зовнішній вигляд"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Зовнішній вигляд</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">перемістити на бічну панель</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">сховати</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Матеріал з Вікіпідручника</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><div class="subpages">< <bdi dir="ltr"><a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8" title="Основні числові системи">Основні числові системи</a></bdi></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="uk" dir="ltr"><div class="infobox sisterproject"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Wikipedia-logo-v2.svg" class="mw-file-description" title="Логотип Вікіпедії"><img alt="Логотип Вікіпедії" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/50px-Wikipedia-logo-v2.svg.png" decoding="async" width="50" height="46" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/75px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/100px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></a><figcaption>Логотип Вікіпедії</figcaption></figure> <a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D0%B5%D0%B4%D1%96%D1%8F" class="mw-redirect" title="Вікіпедія">Вікіпедія</a> має пов'язану з цією темою інформацію на сторінці <a href="https://uk.wikipedia.org/wiki/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" class="extiw" title="w:Цілі числа">Цілі числа</a> </div> У загальному випадку операція віднімання у множині натуральних чисел не виконується, оскільки не кожна пара <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \times \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b25f8c4219e47350185be7ebdc5250c8d59ab35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \times \mathbb {N} }"> натуральних чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7e5710198f33b00695903460983021e75860e2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}"> має образ у <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> за відображення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {N} \times \mathbb {N} \to \mathbb {N} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {N} \times \mathbb {N} \to \mathbb {N} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c46f0d2dca712c9484096013fab121ebf4a5da9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.351ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {N} \times \mathbb {N} \to \mathbb {N} .}"> Іншими словами, якщо <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo><</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a<b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f9e3247a4373970cad2b3b37920af5d23ec7c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.973ex; height:2.509ex;" alt="{\displaystyle a<b,}"> то значення виразу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"> у загальному випадку не належить множині натуральних чисел, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\notin \mathbb {N} ^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>∉</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\notin \mathbb {N} ^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83d177c83f3b86c0822860fc67498426c9ddb49f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.29ex; height:3.176ex;" alt="{\displaystyle (a,b)\notin \mathbb {N} ^{3}.}"> Це стало причиною для збільшення множини натуральних чисел до множини цілих чисел. При розширенні натуральних чисел натуральних чисел до множини цілих <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> чисел необхідно щоб система цілих чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> відповідала певним властивостям по відношенню до системи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1252d3e79f5246bb7650b1beb5c0336cd48d66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.325ex; height:2.509ex;" alt="{\displaystyle \mathbb {N} ,}"> зокрема: <ul><li>система <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"> повинна бути підмножиною <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08441ac98d55673ed5972fc3b92ca6bf9401d1e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.584ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,\,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \subset \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \subset \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/027e3f06a2b7eb019b166d1b2dd46d0914eed3fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.327ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} \subset \mathbb {Z} }">;</li> <li>на системі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> повинні бути визначені операції додавання та множення, причому таким чином, щоб сума й добуток двох натуральних чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> як елементів системи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> співпадали відповідно із сумою <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"> та добутком <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620419d3ed53abc98659a5fc0f3a5eb6177830ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}"> чисел у системі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">;</li> <li>операції додавання та множення, визначені на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}"> повинні мати ті самі властивості (розподільну, сполучну й переставну), що й операції додавання й множення натуральних чисел;</li> <li>у системі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> повинна виконуватися операція віднімання (тобто ми не накладаємо на вираз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"> умову <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a>b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>a</mi> <mo>></mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall a>b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd601c42fe48172ef1ec8b0fba1baa10138a54ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.265ex; height:2.509ex;" alt="{\displaystyle \forall a>b,}"> що читається "для усіх <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> більше <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">").</li></ul> Цілі числа - це натуральні числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"> протилежні до них <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f00139753ecf4fe00a10a17bd5620b70a61b29e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.203ex; height:2.176ex;" alt="{\displaystyle -n}"> та число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e773e0593223c306a3e6852348177d1934962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 0.}"> Вважається, що число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> є протилежним самому собі. Їх можна зобразити на числовій прямій. <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Cili-chislla.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Cili-chislla.tif/lossy-page1-799px-Cili-chislla.tif.jpg" decoding="async" width="799" height="45" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Cili-chislla.tif/lossy-page1-1199px-Cili-chislla.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Cili-chislla.tif/lossy-page1-1598px-Cili-chislla.tif.jpg 2x" data-file-width="799" data-file-height="45" /></a><figcaption></figcaption></figure> Вираз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}"> означає, що записане число протилежне числу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}"> Таким чином, приписавши знак <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}"> до числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 23,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>23</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 23,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00fa1fd0e952159049098ef31e17e9e4d5dbbd77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.972ex; height:2.509ex;" alt="{\displaystyle 23,}"> отримаємо протилежне до нього, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -23.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>23.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -23.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbf84cda57678b36edbf3155381fd6e1624500f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.78ex; height:2.343ex;" alt="{\displaystyle -23.}"> Повторне застосування операції обернення, наприклад <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(-12),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>12</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(-12),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f566567ec0871aec37e9b036b4e30caad495ed5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.397ex; height:2.843ex;" alt="{\displaystyle -(-12),}"> в результаті дає число, протилежне до оберненого числа, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(-12)=12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>12</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(-12)=12.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8495651d00df8435305b64beb3de014302d605c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.821ex; height:2.843ex;" alt="{\displaystyle -(-12)=12.}"> Взагалі, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(-a)=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(-a)=a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1e40f2857656ae1cbfaa94cb8eef81f8f5e05bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.63ex; height:2.843ex;" alt="{\displaystyle -(-a)=a.}"> Варто зауважити, що використання дужок є обов'язковим. Запис виду <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle --a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle --a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7584827cc8dd32aac6d594fe4e4168c47bed8ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.878ex; height:2.176ex;" alt="{\displaystyle --a}"> не вживають. Зрозуміло, що <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(-(-a))=-a,\,\,-(-(-(-a)))=a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(-(-a))=-a,\,\,-(-(-(-a)))=a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9090915f499be0cf794ecd87a31c2c5a79e73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.436ex; height:2.843ex;" alt="{\displaystyle -(-(-a))=-a,\,\,-(-(-(-a)))=a}"> тощо. У виразах, наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25-(3\cdot (-6))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25-(3\cdot (-6))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54b463d27630e5c2079e147130f61418b8913064" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.596ex; height:2.843ex;" alt="{\displaystyle 25-(3\cdot (-6))}">, пишуть <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 25-(-18)=25+18=43.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>25</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>18</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>25</mn> <mo>+</mo> <mn>18</mn> <mo>=</mo> <mn>43.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 25-(-18)=25+18=43.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2fadeea4124601167e4b67bc9ef394c6e2eff9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.767ex; height:2.843ex;" alt="{\displaystyle 25-(-18)=25+18=43.}"> У виразах, де застосовується множення чи ділення, наприклад <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4\cdot (-13)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>13</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\cdot (-13)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9a4a2a4f999c42df1a716ee6b2765d288498f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.784ex; height:2.843ex;" alt="{\displaystyle 4\cdot (-13)}"> чи <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12:(-6),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <mo>:</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12:(-6),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7151cf467088294aba001c8648de8ffc87fb00c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.689ex; height:2.843ex;" alt="{\displaystyle 12:(-6),}"> від'ємне число записується у дужках. При множенні додатного цілого числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> на від'ємне ціле число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-b),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-b),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbb3263d6ea6859d6ffb5de63abd47222cb597c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.262ex; height:2.843ex;" alt="{\displaystyle (-b),}"> добуток буде від'ємним цілим числом; якщо від'ємне <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a481c59f1c92c692cfcb4320af4ee08bf7a925af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.847ex; height:2.843ex;" alt="{\displaystyle (-a)}"> число множиться на від'ємне число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48827b58299886483952c6fcf3f8f55948ab2fc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.615ex; height:2.843ex;" alt="{\displaystyle (-b)}">, то добуток буде додатним числом: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)\cdot (-b)=-(-(a\cdot b))=a\cdot b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)\cdot (-b)=-(-(a\cdot b))=a\cdot b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0701a411cf2f8fedb788ccf79a548740d1e0edb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.033ex; height:2.843ex;" alt="{\displaystyle (-a)\cdot (-b)=-(-(a\cdot b))=a\cdot b.}">. Добуток трьох від'ємних чисел - від'ємне число, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-a)\cdot (-b)\cdot (-c)=-(-(-(a\cdot b\cdot c)))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-a)\cdot (-b)\cdot (-c)=-(-(-(a\cdot b\cdot c)))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6075a66f61fd3976a5d15ee3c822646bea73592e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.988ex; height:2.843ex;" alt="{\displaystyle (-a)\cdot (-b)\cdot (-c)=-(-(-(a\cdot b\cdot c)))}"> (тут ми винесли мінус за дужки) тощо. Операція ділення у загальному випадку не виконується у системі цілих чисел, оскільки, наприклад, значення виразу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5:2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>:</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5:2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d194324ba1eb86220ce5881a2d54709fad48adc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.262ex; height:2.176ex;" alt="{\displaystyle 5:2}"> не належить множині <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}"> Тобто число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> не ділиться націло на число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}"> Так само число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\,071}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mspace width="thinmathspace" /> <mn>071</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\,071}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20b14e0a18869992d7632235cc44a4a730f391a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.037ex; height:2.176ex;" alt="{\displaystyle 1\,071}"> не ділиться націло на число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> тощо. Таким чином, вважаючи змінну <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> цілим числом, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad6b3baee714b61d6caa3a6edede2d2c6cf14f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.078ex; height:2.509ex;" alt="{\displaystyle m\in \mathbb {Z} ,}"> ділення довільного цілого числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff96cba8be0fa52dd0b472c9f40e7d6a3cb75db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.721ex; height:2.176ex;" alt="{\displaystyle x\in \mathbb {Z} }"> на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dad66d19bb37bc69223cb004be2ea5dd95f9564c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.687ex; height:2.009ex;" alt="{\displaystyle m,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>:</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x:m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b76122c54ff9d010de500b7062d0c4855d69128a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.307ex; height:1.676ex;" alt="{\displaystyle x:m}"> або <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {x}{m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mi>m</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {x}{m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4db19ab4f479dbd113293fce350afd27c269adab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.523ex; height:4.676ex;" alt="{\displaystyle {\frac {x}{m}},}"> вважається не визначеним в загальному випадку (або частково (тобто не усюди на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">) визначеним у сенсі часткового відображення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {Z} \times \mathbb {Z} \to \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\mathbb {Z} \times \mathbb {Z} \to \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bc13ecb26f86883b6f38528b9f30956720faf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.321ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {Z} \times \mathbb {Z} \to \mathbb {Z} }">). <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Абсолютна_величина">Абсолютна величина</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=1" title="Редагувати розділ: Абсолютна величина">ред.</a>]</div> Абсолютною величиною цілого числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> називається невід'ємне з чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e0982b5868a66be1ed3ad7ef4bcd3d3db20f982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:3.038ex; height:2.176ex;" alt="{\displaystyle -a}"> і позначається <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a9e4424efba80112e20c4251661c2e59a85faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.17ex; height:2.843ex;" alt="{\displaystyle |a|.}"> Наприклад, абсолютною величиною числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -13}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>13</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -13}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4535a0d4479ed2ed4228d61fa0633637c525aa98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -13}"> є число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-13|=-(-13)=13.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>13</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>13</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>13.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |-13|=-(-13)=13.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07f3122367cfbd2fe0dfcbd0e73b18b9adf5ea05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.378ex; height:2.843ex;" alt="{\displaystyle |-13|=-(-13)=13.}"> Таким чином, модуль додатного числа є число додатне, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |a|=a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |a|=a,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/224a6ae26db266aac10e20243fe1510b966a1a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.499ex; height:2.843ex;" alt="{\displaystyle |a|=a,}"> а модуль від'ємного числа - число, протилежне йому, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-a|=-(-a)=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |-a|=-(-a)=a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/743c535385cc8ac38ca48cd51cfd29893e715859" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.093ex; height:2.843ex;" alt="{\displaystyle |-a|=-(-a)=a.}">. Зрозуміло, що модуль набуває лише невід'ємних значень. Вважається, що модуль числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> дорівнює нулю, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0|=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bf0a87d4421e4c322fed626bb68e613c2a9ff41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.364ex; height:2.843ex;" alt="{\displaystyle |0|=0.}"> Геометричний зміст поняття абсолютної величини полягає у тому, що абсолютна величина означає відстань від початку відліку до відзначеної точки, яка позначає відповідне число на координатній прямій. Тобто на скільки одиниць даної числової прямої віддалена дана точка від початку відліку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1cdb120fda7ea217dfd4d6b9e297a863f45de87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.42ex; height:2.176ex;" alt="{\displaystyle O.}"> Позначмо на числовій прямій точки <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=-5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mo>−</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=-5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c49cb7301d57150fe288b03b60570840825462f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.652ex; height:2.343ex;" alt="{\displaystyle L=-5}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5138ef25b1971cdd3519ae83ec5b15d7adea176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.674ex; height:2.176ex;" alt="{\displaystyle C=3.}"> Точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> віддалена на 5 одиниць від початку відліку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0639603a009660b47d9e62682f8cf79d20edcecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle O,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |-5|=-(-5)=5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |-5|=-(-5)=5.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0af8dee9513fd3ab13d98b7d6fe7f6f0111bf05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.891ex; height:2.843ex;" alt="{\displaystyle |-5|=-(-5)=5.}"> Точка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"> знаходиться на відстані 3 одиниці від точки початку відліку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0639603a009660b47d9e62682f8cf79d20edcecb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.42ex; height:2.509ex;" alt="{\displaystyle O,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |3|=3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |3|=3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63185efc0d8252bfe73511ba9caad96be32022b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.364ex; height:2.843ex;" alt="{\displaystyle |3|=3.}"> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Abs53.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Abs53.tif/lossy-page1-649px-Abs53.tif.jpg" decoding="async" width="649" height="115" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/62/Abs53.tif/lossy-page1-974px-Abs53.tif.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/62/Abs53.tif/lossy-page1-1298px-Abs53.tif.jpg 2x" data-file-width="649" data-file-height="115" /></a><figcaption></figcaption></figure> <div class="mw-heading mw-heading2"><h2 id="Цілі_вирази">Цілі вирази</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=2" title="Редагувати розділ: Цілі вирази">ред.</a>]</div> Розгляньмо дві групи виразів: першу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b,\quad \quad 7-x,\quad \quad {\frac {1}{3}}\cdot b,\quad \quad a-{\frac {b}{2}},\quad \quad {\frac {a+b}{12}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>7</mn> <mo>−</mo> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>⋅</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mrow> <mn>12</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b,\quad \quad 7-x,\quad \quad {\frac {1}{3}}\cdot b,\quad \quad a-{\frac {b}{2}},\quad \quad {\frac {a+b}{12}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20ab9144488410c14a9696cf0c83c0b65a5725c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.25ex; height:5.343ex;" alt="{\displaystyle a\cdot b,\quad \quad 7-x,\quad \quad {\frac {1}{3}}\cdot b,\quad \quad a-{\frac {b}{2}},\quad \quad {\frac {a+b}{12}},}"> та другу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {12}{a}},\quad \quad {\frac {a}{b}},\quad \quad 12:c,\quad \quad {\frac {6+a}{2\cdot b}}\cdot (-x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>12</mn> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <mi>b</mi> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>12</mn> <mo>:</mo> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mo>+</mo> <mi>a</mi> </mrow> <mrow> <mn>2</mn> <mo>⋅</mo> <mi>b</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {12}{a}},\quad \quad {\frac {a}{b}},\quad \quad 12:c,\quad \quad {\frac {6+a}{2\cdot b}}\cdot (-x).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d84cd7ce71aa62a3d5ca38568c5c72e9af0b4026" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.875ex; height:5.343ex;" alt="{\displaystyle {\frac {12}{a}},\quad \quad {\frac {a}{b}},\quad \quad 12:c,\quad \quad {\frac {6+a}{2\cdot b}}\cdot (-x).}"> Перша група виразів не містить дії ділення на вираз із змінними. Такі вирази називаються цілими. Вирази ж другої групи містять дію ділення на вираз зі змінними і називаються дробовими виразами. У виразах дію множення часто не вживають, і замість <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot k-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot k-2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996ebbc9229ef67dffb4868d17246327c1b966b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.056ex; height:2.343ex;" alt="{\displaystyle 5\cdot k-2}"> пишуть просто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5k-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>k</mi> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5k-2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4aaafe2d5b9b2b762cd62621ea18e66d930a584c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.023ex; height:2.343ex;" alt="{\displaystyle 5k-2.}"> Таким самим чином перед дужками знак дії множення не вживається, наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(b+c).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b+c).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7510c367a1f74da437537c3d753beebfaa63919c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.531ex; height:2.843ex;" alt="{\displaystyle a(b+c).}"> У подальшому знак дії множення перестане вживатися у подібних випадках. Як згадувалося на початку, вирази зі змінними використовуються для запису формул. Наприклад, формулами можна задати цілі числа, що діляться на задане натуральне число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> і дають одну і ту саму остачу. Якщо кожне з чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> ділиться націло на число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"> то й сума <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"> також ділиться націло на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"> Якщо ж число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> ділиться націло на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"> а число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> не ділиться на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae5e8f9eb465084d3a00a24026b80652b74ef58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.654ex; height:2.009ex;" alt="{\displaystyle c,}"> то й сума <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2391acf09244b9dba74eb940e871a6be7e7973a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a+b}"> не ділиться на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"> Разом із діленням націло розглядають ділення із остачею. Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21:9=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> <mo>:</mo> <mn>9</mn> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 21:9=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4bfef170c10f13db685c67dd615f44dac02c270" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.685ex; height:2.176ex;" alt="{\displaystyle 21:9=2}"> із остачею 3. Таким чином, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a-r):b=k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>b</mi> <mo>=</mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-r):b=k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ff32a6c5a5eced74927ce7d953f7e66556dc7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.819ex; height:2.843ex;" alt="{\displaystyle (a-r):b=k,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> - остача, а <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> - неповна частка від ділення. У нашому прикладі, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (21-{\overset {r}{3}}):9={\overset {k}{2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>21</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mi>r</mi> </mover> </mrow> <mo stretchy="false">)</mo> <mo>:</mo> <mn>9</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mi>k</mi> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (21-{\overset {r}{3}}):9={\overset {k}{2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3be34ad2d06b0f55160fb684be084906d8a2e2ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.144ex; height:4.509ex;" alt="{\displaystyle (21-{\overset {r}{3}}):9={\overset {k}{2}}.}"> Таким чином, остача <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2452d01ccc3bc930e3e2bff9fb18d2a425272ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=3}">, неповна частка <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090a4d3b2c87ca76bab993bdfc89426e26a7a509" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.119ex; height:2.176ex;" alt="{\displaystyle k=2.}"> Взагалі, щоб розділити ціле число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> на натуральне число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> із остачею, потрібно знайти такі цілі числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e185ab9c990830d5055fa3ae698a4225ce67e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle k,}"> щоб справджувалася рівність <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=n\cdot k+r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>⋅</mo> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n\cdot k+r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0b13fa7faf0858410326a5c615b59424a67b73e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.96ex; height:2.509ex;" alt="{\displaystyle m=n\cdot k+r,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r\leq n-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r\leq n-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a54cf1582c81a653de85b082fe3afe700a9e888a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.452ex; height:2.343ex;" alt="{\displaystyle 0\leq r\leq n-1.}"> Остач від ділення цілих чисел на натуральне число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> може бути <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n:\,0,1,2,...,n-2,n-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>:</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n:\,0,1,2,...,n-2,n-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f893a428aa21f8718c6fc9e7770c4c4b5c7088e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.92ex; height:2.509ex;" alt="{\displaystyle n:\,0,1,2,...,n-2,n-1.}"> Наприклад, знайдемо остачу від ділення числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c81bb3362813dd9edacb8e6d6552373a561dd2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -17}"> на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75e17f7e7094c457fa7c8f743c3d8622eebd8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 3.}"> Для цього запишемо число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c81bb3362813dd9edacb8e6d6552373a561dd2c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.133ex; height:2.343ex;" alt="{\displaystyle -17}"> у вигляді <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -17=3k+r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>17</mn> <mo>=</mo> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -17=3k+r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9daa2fedd2fa62dfba83296c64018ccf1fa4b4af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.141ex; height:2.509ex;" alt="{\displaystyle -17=3k+r,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> - цілі числа і <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r\leq 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r\leq 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09436a6a3bb653c34ee73852a33bdf3c24f3573f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.217ex; height:2.343ex;" alt="{\displaystyle 0\leq r\leq 2.}"> Щоб <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}"> знаходилося у межах від <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993d53082bc05c266933af9a892e1ce2128547cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 2,}"> потрібно взяти <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=-6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=-6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f731f162b63f76deaabf6be196dfa4830676730b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.927ex; height:2.343ex;" alt="{\displaystyle k=-6.}"> Таким чином, за <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6584ba3b7843583b757896c2f0686efc0489e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.31ex; height:2.176ex;" alt="{\displaystyle r=1}"> справджується рівність <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17=3\cdot (-6)+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17=3\cdot (-6)+1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6778ac114917507ede3f96c3a13c6ddd2c129271" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.695ex; height:2.843ex;" alt="{\displaystyle 17=3\cdot (-6)+1.}"> Нехай <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> - відрізки і відрізок <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> є довшим за відрізок <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>></mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a>b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a>b}"> (або <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b<a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo><</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b<a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec536027666ac731c00e6db3a3b0b10d7ebbe031" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle b<a}">). Вкладімо відрізок <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> до відрізку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> стільки разів, скільки можливо; отримаємо остачу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d823e1fe9c4338095d7570804e11b9eaef344567" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.009ex;" alt="{\displaystyle r_{1}.}"> На наступному малюнку видно, що відрізок довжиною <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> може вмістити <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> відрізків довжиною <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c20044cce355e2612a7d7a7bd938b146b4e79ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b;}"> таким чином, можна записати <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=5b+r_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>5</mn> <mi>b</mi> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=5b+r_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb65ac7da71edace4226f36ba1f00ad209a983a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.078ex; height:2.509ex;" alt="{\displaystyle a=5b+r_{1}.}"> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Immersionbsegment.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Immersionbsegment.tif/lossless-page1-387px-Immersionbsegment.tif.png" decoding="async" width="387" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Immersionbsegment.tif/lossless-page1-581px-Immersionbsegment.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Immersionbsegment.tif/lossless-page1-774px-Immersionbsegment.tif.png 2x" data-file-width="387" data-file-height="145" /></a><figcaption></figcaption></figure> Далі, вкладімо відрізок остачі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea214f2b31fb3869344bb9311da41c5cc38a99e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{1}}"> до відрізку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> стільки разів, скільки можливо (по аналогії із вкладанням <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">); зокрема, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2r_{1}+r_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2r_{1}+r_{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f60598e0d3e3345fc81b337d6a808e3805bd3e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.951ex; height:2.509ex;" alt="{\displaystyle b=2r_{1}+r_{2}.}"> Утвориться остача <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1bcfd985dbd4224606059eb45e05c9b493b090c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.009ex;" alt="{\displaystyle r_{2}.}"> <figure class="mw-default-size mw-halign-center" typeof="mw:File"><a href="/wiki/%D0%A4%D0%B0%D0%B9%D0%BB:Immersionbsegment12.tif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Immersionbsegment12.tif/lossless-page1-79px-Immersionbsegment12.tif.png" decoding="async" width="79" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Immersionbsegment12.tif/lossless-page1-119px-Immersionbsegment12.tif.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Immersionbsegment12.tif/lossless-page1-158px-Immersionbsegment12.tif.png 2x" data-file-width="79" data-file-height="120" /></a><figcaption></figcaption></figure> Аналогічним чином вкладімо послідовно відрізки <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"> до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed9d9537b1d764e5c4f986e3c63f3bfb362df6c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.75ex; height:2.009ex;" alt="{\displaystyle r_{1},}"> відрізки <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fc5930cbb780220b209b444707ad9e2ba82c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{3}}"> до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"> тощо. Якщо, відкладаючи черговий відрізок <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"> до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d05e41cff8db5ef319b4c9d35f41901ad1118d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.014ex; height:2.009ex;" alt="{\displaystyle r_{n-1},}"> ми не отримаємо остачі (точніше, отримаємо нульову остачу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n+1}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n+1}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7669c4c05cf68a5746e83c2934e388911ee33c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.629ex; height:2.509ex;" alt="{\displaystyle r_{n+1}=0}">), то відрізок довжини <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"> є найбільшою спільною мірою відрізків довжин <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef051eb30c89e5493d672f6479566c673b0890a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.644ex; height:2.176ex;" alt="{\displaystyle b.}"> Якщо <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> є цілими, то усі остачі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1},r_{2},...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},r_{2},...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29c9d889b7e22d1c0ca7870883e9f5458568e42f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.988ex; height:2.009ex;" alt="{\displaystyle r_{1},r_{2},...}"> також цілі невід'ємні. В силу лінійного порядку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b>r_{1}>r_{2}>...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b>r_{1}>r_{2}>...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15e224cf34c7bd27f7687fad6d209cf0f8935c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.568ex; height:2.509ex;" alt="{\displaystyle b>r_{1}>r_{2}>...}"> (або, що те саме, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\supset r_{1}\supset r_{2}\supset ...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>⊃</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊃</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊃</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\supset r_{1}\supset r_{2}\supset ...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/877bb1bded674d765500c846938309c923a4e366" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.568ex; height:2.509ex;" alt="{\displaystyle b\supset r_{1}\supset r_{2}\supset ...}">) процес рано чи пізно закінчиться. Останнє ненульове число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57784cdf7f49f2c46baad4eb7b6a7d5c14eb5fa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.267ex; height:2.009ex;" alt="{\displaystyle r_{n}}"> є найбільшим спільним дільником чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{n}={\text{НСД}}(a,b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{n}={\text{НСД}}(a,b).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed901909abc99f833ac3e0fb43e128b8ae4c472" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.333ex; height:3.343ex;" alt="{\displaystyle r_{n}={\text{НСД}}(a,b).}"> <div class="mw-heading mw-heading2"><h2 id="Метод_порівняння">Метод порівняння</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=3" title="Редагувати розділ: Метод порівняння">ред.</a>]</div> Два числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> при діленні на натуральне число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> дають один і той самий залишок <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/250644a0f511e9078be6f89ba78a606a0e08c0a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.695ex; height:2.009ex;" alt="{\displaystyle r,}"> у питаннях подільності по відношенню до <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> мають ряд однакових властивостей. Зокрема, будь-який спільний дільник чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> буде одночасно й спільним дільником для <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> Зокрема, найбільший спільний дільник для пари <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9825dc79e27764c957cdcfdf4c97cf26d80983" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.468ex; height:2.843ex;" alt="{\displaystyle (a,n)}"> чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> співпадає із найбільшим спільним дільником <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b,n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a9f931d43fba22446f4002b9e4bea6fd31420fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.236ex; height:2.843ex;" alt="{\displaystyle (b,n)}"> чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> Справді, якщо <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=21}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>21</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=21}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c968d94cd2c31a0a1fe2dd7245fd9e47d5228f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.653ex; height:2.176ex;" alt="{\displaystyle a=21}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=33}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>33</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=33}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04d22e287f1a8b1589700bde4beb9b23991d5e0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.421ex; height:2.176ex;" alt="{\displaystyle b=33}"> дають при діленні на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0365f0b9f2721ed3ebb488a96d7348d978acf8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=6}"> одну і ту саму остачу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3fa74ab9cadb9d08c114d19205c76a5c10aedb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.956ex; height:2.509ex;" alt="{\displaystyle r=3,}"> то <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=3\cdot n+3,\quad \quad b=5\cdot n+3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=3\cdot n+3,\quad \quad b=5\cdot n+3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/005d3f55d215eb17190b1f400302bbb5a06eb359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.228ex; height:2.509ex;" alt="{\displaystyle a=3\cdot n+3,\quad \quad b=5\cdot n+3,}"> тому різниця <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b=(3-5)\cdot n=-12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b=(3-5)\cdot n=-12.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c46d190564f4d0a32585cab869e95a425f887ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.093ex; height:2.843ex;" alt="{\displaystyle a-b=(3-5)\cdot n=-12.}"> ділиться на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> Число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"> є неповною часткою при діленні <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 21}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>21</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 21}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77ebb9ccf6080ba5c9a6ea8973cb2f26c50211cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 21}"> на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90619051dad04a036539bb64306c01dca48840d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 6,}"> а число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> є неповною часткою при діленні <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 33}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>33</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 33}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7069e8371d2801429050dd3c82807e391beaed01" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 33}"> на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641abae36403783c71568aae30295e0e32b72daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 6.}"> Відтак, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=(3-5)\cdot n+b,\quad \quad b=a-(3-5)\cdot n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> <mo>+</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=(3-5)\cdot n+b,\quad \quad b=a-(3-5)\cdot n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9df810513319997104070a29b7412a62ed3871c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.755ex; height:2.843ex;" alt="{\displaystyle a=(3-5)\cdot n+b,\quad \quad b=a-(3-5)\cdot n.}"> Числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> прийнято називати порівнюваними за модулем <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"> якщо обидва числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> дають однакову остачу при діленні на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> Це позначається наступним чином: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\equiv b(\mod n).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≡</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mspace width="1em" /> <mi>mod</mi> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\equiv b(\mod n).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13979fd2e7d03310f9245dd158eacb735ba1c995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.051ex; height:2.843ex;" alt="{\displaystyle a\equiv b(\mod n).}"> При визначенні порівнюваності двох чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> вимога рівності остач від ділення на спільний дільник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> може бути замінена рівносильною, але більш зручною для перевірки вимогою щоб різниця двох даних чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b80866c2bf2f1bc1f2e4c97e7937f5663150ea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.068ex; height:2.343ex;" alt="{\displaystyle a-b}"> ділилася на модуль <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> У нашому прикладі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (21-33):6=-2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>21</mn> <mo>−</mo> <mn>33</mn> <mo stretchy="false">)</mo> <mo>:</mo> <mn>6</mn> <mo>=</mo> <mo>−</mo> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (21-33):6=-2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8bd98ec7ef2e10b1d4ca82d74b111476a632a8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.115ex; height:2.843ex;" alt="{\displaystyle (21-33):6=-2.}"> Цілі числа при діленні на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}"> можуть давати в остачі лише <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,\,\,1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,\,\,1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03ece3b5d56c50950a788ab177d3744a963d44d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.133ex; height:2.509ex;" alt="{\displaystyle 0,\,\,1}"> або <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}"> Відповідно, цілі числа розшаровуються на 3 класи по модулю <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/991e33c6e207b12546f15bdfee8b5726eafbbb2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 3}">: <table class="wikitable"> <tbody><tr> <th>Цілі числа</th> <th>Остача при діленні на 3</th> <th>Вираз </th></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ...,-9,-6,-3,0,3,6,9,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>9</mn> <mo>,</mo> <mo>−</mo> <mn>6</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ...,-9,-6,-3,0,3,6,9,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c848e42c2e6bb8ac50eb47b3b26d7d0c1382d84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.65ex; height:2.509ex;" alt="{\displaystyle ...,-9,-6,-3,0,3,6,9,...}"></td> <td>0</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca782a42fc66d7dd90d38cecd534a3977fc8af9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.374ex; height:2.176ex;" alt="{\displaystyle 3k}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ...,-8,-5,-2,1,4,7,10,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>8</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ...,-8,-5,-2,1,4,7,10,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7def881ba121542bbb0dc1778c2c11b404d4d7a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.812ex; height:2.509ex;" alt="{\displaystyle ...,-8,-5,-2,1,4,7,10,...}"></td> <td>1</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3k+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3k+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a6128d1598e2bdb1907ec6b40269511683d1ebc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.377ex; height:2.343ex;" alt="{\displaystyle 3k+1}"> </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ...,-7-4,-1,2,5,8,11,...}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo>−</mo> <mn>7</mn> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ...,-7-4,-1,2,5,8,11,...}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f32d75989d7f2df37c7c87afd31cd34afcf54f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.81ex; height:2.509ex;" alt="{\displaystyle ...,-7-4,-1,2,5,8,11,...}"></td> <td>2</td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3k+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3k+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8c6865e58b7169bdece2a525777da6a00d6b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.377ex; height:2.343ex;" alt="{\displaystyle 3k+2}"> </td></tr></tbody></table> Про числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m=3\cdot k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=3\cdot k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33c0b64bf9a6020ae529e3b29c1b9efaa4e98de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.192ex; height:2.176ex;" alt="{\displaystyle m=3\cdot k}"> говорять, що вони націло діляться на число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75e17f7e7094c457fa7c8f743c3d8622eebd8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 3.}"> Взагалі, число класів по модулю <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397bfafc701afdf14c2743278a097f6f2957eabb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.042ex; height:2.009ex;" alt="{\displaystyle n,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,1,2,3,...,n-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,1,2,3,...,n-1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0cd22d6c43d224fa4101c2e74ab56490f60d9c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.966ex; height:2.509ex;" alt="{\displaystyle 0,1,2,3,...,n-1.}"> Клас, що характеризується даною остачею <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r\leq n-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>≤</mo> <mi>r</mi> <mo>≤</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\leq r\leq n-1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7382b8db40394aa0480b9462e150b7d2eb38af8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.452ex; height:2.509ex;" alt="{\displaystyle 0\leq r\leq n-1,}"> утворюють числа виду <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot k+r,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>⋅</mo> <mi>k</mi> <mo>+</mo> <mi>r</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\cdot k+r,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed0c320ce1ad27de9c9c97c1f18e3dc402e5f122" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.821ex; height:2.509ex;" alt="{\displaystyle n\cdot k+r,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> - будь-яке ціле число. <div class="mw-heading mw-heading2"><h2 id="Тотожні_вирази">Тотожні вирази</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=4" title="Редагувати розділ: Тотожні вирази">ред.</a>]</div> Вирази називаються тотожними, якщо за будь-яких значень змінних, які до них входять, значення цих виразів дорівнюють одне одному. Наприклад, нехай маємо вирази <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x-5y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x-5y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2eff9712305bf23bacb751ec97993833c87495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.65ex; height:2.509ex;" alt="{\displaystyle 5x-5y}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5(x-y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5(x-y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e21c4d975da677d8c4115ea5be531d24b0ec399" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.944ex; height:2.843ex;" alt="{\displaystyle 5(x-y).}"> Якщо підставити значення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d366942ea55e335b92439f564940a422e749648" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.491ex; height:2.176ex;" alt="{\displaystyle a=4}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1c4ed83cc30f61cb9992767029f9b4653237ee5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.905ex; height:2.509ex;" alt="{\displaystyle b=2,}"> то отримаємо рівності <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x-5y=5\cdot 4-5\cdot 2=20-10=10;\quad \quad 5(x-y)=5x-5y=5\cdot 4-5\cdot 2=10.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mn>4</mn> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>20</mn> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>10</mn> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mn>4</mn> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>10.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x-5y=5\cdot 4-5\cdot 2=20-10=10;\quad \quad 5(x-y)=5x-5y=5\cdot 4-5\cdot 2=10.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba2917040aa49072b71aa657270934b8f6cbf11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.352ex; height:2.843ex;" alt="{\displaystyle 5x-5y=5\cdot 4-5\cdot 2=20-10=10;\quad \quad 5(x-y)=5x-5y=5\cdot 4-5\cdot 2=10.}"> У цьому випадку кажуть, що два вирази <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x-5y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x-5y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2eff9712305bf23bacb751ec97993833c87495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.65ex; height:2.509ex;" alt="{\displaystyle 5x-5y}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5(x-y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97ce811826e240d21037d8ae355595a6c7e7be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.297ex; height:2.843ex;" alt="{\displaystyle 5(x-y)}"> є тотожно рівними. Якщо два тотожно рівні вирази <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x-5y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x-5y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2eff9712305bf23bacb751ec97993833c87495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.65ex; height:2.509ex;" alt="{\displaystyle 5x-5y}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5(x-y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5(x-y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97ce811826e240d21037d8ae355595a6c7e7be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.297ex; height:2.843ex;" alt="{\displaystyle 5(x-y)}"> сполучити відношенням рівності <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c1d352943e2089b5de03406b09a2239471050a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:1.843ex;" alt="{\displaystyle =,}"> то отримаємо рівність <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x-5y=5(x-y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>=</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x-5y=5(x-y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6cbac0d3f2596bdcc0875ffe53616b490f2b4c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.693ex; height:2.843ex;" alt="{\displaystyle 5x-5y=5(x-y),}"> яка справджується за будь-яких значень <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.802ex; height:2.009ex;" alt="{\displaystyle y.}"> Така рівність виразів називається тотожністю. Прикладами тотожностей, перш за все, є закони, які виражають властивості операцій додавання та множення чисел: <ul><li>переставна властивість: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+b=b+a;\quad \quad ab=ba;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+b=b+a;\quad \quad ab=ba;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/190d03efc212670d20b58d894b36677e13508dce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.113ex; height:2.509ex;" alt="{\displaystyle a+b=b+a;\quad \quad ab=ba;}"></li> <li>сполучна властивість: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)=(a+b)+c;\quad \quad (ab)c=a(bc);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>c</mi> <mo>;</mo> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)=(a+b)+c;\quad \quad (ab)c=a(bc);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b875a680bfb18f6a9a9b4100097d6e313bf4de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.058ex; height:2.843ex;" alt="{\displaystyle a+(b+c)=(a+b)+c;\quad \quad (ab)c=a(bc);}"></li> <li>розподільна властивість: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(b+c)=ab+ac.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b+c)=ab+ac.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671d624300064920a8d7a1ae9cc11b397f4b83ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.934ex; height:2.843ex;" alt="{\displaystyle a(b+c)=ab+ac.}"></li></ul> Правила розкриття дужок також виражаються тотожностями: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(b+c)=a+b+c,\quad \quad a-(b+c)=a-b-c,\quad \quad a-(b-c)=a-b+c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(b+c)=a+b+c,\quad \quad a-(b+c)=a-b-c,\quad \quad a-(b-c)=a-b+c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd79437f2a33a4b70203d311961a66d6609fbc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:80.218ex; height:2.843ex;" alt="{\displaystyle a+(b+c)=a+b+c,\quad \quad a-(b+c)=a-b-c,\quad \quad a-(b-c)=a-b+c.}"> Зрозуміло при цьому, що якщо перед дужками стоїть знак <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e7363422f6855d1910aeb4922a790ea855288a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.343ex;" alt="{\displaystyle -,}"> то дужки розкриваються зі зміною знаків доданків на протилежні. Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(a-b)=-a+b=b-a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>−</mo> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(a-b)=-a+b=b-a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0037961a50e7c331516b2331b943fc115f4150e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.826ex; height:2.843ex;" alt="{\displaystyle -(a-b)=-a+b=b-a}"> або <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(a+b)=-a-b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(a+b)=-a-b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20fd670766dee179a2c84611600f6f28426b752a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.306ex; height:2.843ex;" alt="{\displaystyle -(a+b)=-a-b.}"> Тотожностями є й наступні рівності: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a+(-b)=a-b,\quad \quad a\cdot (-b)=-ab,\quad \quad (-a)\cdot (-b)=ab,\quad \quad x+0=x,\quad \quad a+(-a)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>x</mi> <mo>+</mo> <mn>0</mn> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a+(-b)=a-b,\quad \quad a\cdot (-b)=-ab,\quad \quad (-a)\cdot (-b)=ab,\quad \quad x+0=x,\quad \quad a+(-a)=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a726cc5d458d4b96b172fdea1489a206f73f8104" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:94.279ex; height:2.843ex;" alt="{\displaystyle a+(-b)=a-b,\quad \quad a\cdot (-b)=-ab,\quad \quad (-a)\cdot (-b)=ab,\quad \quad x+0=x,\quad \quad a+(-a)=0,}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot 0=0,\quad \quad a\cdot 1=a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mn>0</mn> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot 0=0,\quad \quad a\cdot 1=a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5558370d102c314b37067b94fd8403ca61f9277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.058ex; height:2.509ex;" alt="{\displaystyle a\cdot 0=0,\quad \quad a\cdot 1=a.}"> Один вираз можна замінити іншим, більш коротким, якщо звести подібні доданки. Наприклад, у виразі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x-3+5x+12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x-3+5x+12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f06cdcb09d2a1063e9dc09d3a536c842bc21581" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.993ex; height:2.343ex;" alt="{\displaystyle 4x-3+5x+12}"> подібні доданки - це <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51fb5b15dfeeb171680fea02842c438db9395293" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.492ex; height:2.176ex;" alt="{\displaystyle 4x}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mi>x</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9313a32142c8949da618ca357dd055d4680b1800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.139ex; height:2.509ex;" alt="{\displaystyle 5x,}"> що утворюють першу групу подібних доданків; другу групу подібних доданків утворюють числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e8ffb51bb3837cabdd202845d15aedc67a4ac88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -3}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ee8bf1c252cecdcbd35ffeae377d0e64dd73e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 12.}"> Вести подібні доданки - значить скоротити вираз, здійснивши операції над усіма групами доданків, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x+5x=9x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>=</mo> <mn>9</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x+5x=9x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe34806ad006fe079def7fa5afdf8a90ff6418e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.415ex; height:2.343ex;" alt="{\displaystyle 4x+5x=9x}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3+12=9.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>12</mn> <mo>=</mo> <mn>9.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3+12=9.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c9ebc3314d4b615d03edb045f9a813e47cf6c92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.044ex; height:2.343ex;" alt="{\displaystyle -3+12=9.}"> В результаті ми отримуємо короткий вираз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9x+9,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mi>x</mi> <mo>+</mo> <mn>9</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9x+9,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06a6b52d91d611bcbd25c86cc30aec12bcad98b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.142ex; height:2.509ex;" alt="{\displaystyle 9x+9,}"> який є тотожно рівним виразу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x-3+5x+12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x-3+5x+12.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40431ca59bcbca8bcf89c69031844feefb5aec48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.64ex; height:2.343ex;" alt="{\displaystyle 4x-3+5x+12.}"> Така процедура називається спрощенням виразу. Ще раз, спростимо вираз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7x+4+2(-4x+1)=7x-8x+4+2=6-x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>7</mn> <mi>x</mi> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>6</mn> <mo>−</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7x+4+2(-4x+1)=7x-8x+4+2=6-x.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae100c768c449efa3efd1cf63e9d81fdf411dce9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.617ex; height:2.843ex;" alt="{\displaystyle 7x+4+2(-4x+1)=7x-8x+4+2=6-x.}"> Тотожні перетворення використовуються для доведення тотожностей. Щоб довести тотожність, можна скористатися одним із наступних способів: <ul><li>ліву частину відносно знаку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> шляхом тотожних перетворень звести до правої;</li> <li>праву частину відносно знаку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle =}"> звести до лівої;</li> <li>обидві частини (ліву та праву) звести до одного і того ж виразу;</li> <li>відняти від лівої частини праву і запевнитися, що різниця дорівнює нулю.</li></ul> У рівності ліва частина позначена червоним кольором, а права - синім: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=}\color {blue}{-3a-10}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> </mrow> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=}\color {blue}{-3a-10}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/422557f12240f388658f25005e373a6760d18c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.098ex; height:2.843ex;" alt="{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=}\color {blue}{-3a-10}.}"> Перетворимо ліву частину цієї рівності: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=a-3-4a-7=a-4a-3-7=}\color {blue}{-3a-10}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mo>−</mo> <mn>7</mn> <mo>=</mo> <mi>a</mi> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo>−</mo> <mn>7</mn> <mo>=</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> </mrow> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=a-3-4a-7=a-4a-3-7=}\color {blue}{-3a-10}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b88eb2772014a7777adccfc946dffe95513c4c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.231ex; height:2.843ex;" alt="{\displaystyle \color {red}{a-3-(4a+7)}\color {black}{=a-3-4a-7=a-4a-3-7=}\color {blue}{-3a-10}.}"> Таким чином, права і ліва частини співпадають і є тотожностями. При перенесенні виразів та їх частин з однієї частини рівності до іншої, вони змінюють свої знаки. У рівності <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3a-10=-3a-10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3a-10=-3a-10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8fdea632211cb9a7e8ff12eeba148007e9174f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.83ex; height:2.343ex;" alt="{\displaystyle -3a-10=-3a-10}"> перенесемо вираз праворуч відносно рівності у ліву частину: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3a-10-(-3a-10)=-3a+3a-10+10=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>10</mn> <mo>+</mo> <mn>10</mn> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3a-10-(-3a-10)=-3a+3a-10+10=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78fa0217bcdef16d2093f990b84daa6a0d4148ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.151ex; height:2.843ex;" alt="{\displaystyle -3a-10-(-3a-10)=-3a+3a-10+10=0.}"> Таким чином, тотожність доведена. <div class="mw-heading mw-heading2"><h2 id="Степінь_числа_з_натуральним_показником"><span id=".D0.A1.D1.82.D0.B5.D0.BF.D1.96.D0.BD.D1.8C_.D1.87.D0.B8.D1.81.D0.BB.D0.B0_.D0.B7_.D0.BD.D0.B0.D1.82.D1.83.D1.80.D0.B0.D0.BB.D1.8C.D0.BD.D0.B8.D0.BC_.D0.BF.D0.BE.D0.BA.D0.B0.D0.B7.D0.BD.D0.B8.D0.BA.D0.BE.D0.BC">Степінь числа з натуральним показником</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=5" title="Редагувати розділ: Степінь числа з натуральним показником">ред.</a>]</div> Степінь числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> з натуральним показником <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> - це добуток, що складається з <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> множників <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f059f053fcf9f421b7c74362cf3bd5ed024e19d1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{множників}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>множників</mtext> </mrow> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{множників}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c18759be8625dee4e7154d619a1bfd54d377a991" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:16.583ex; height:6.676ex;" alt="{\displaystyle a^{n}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{множників}}}.}"> Читається: степінь <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> з основою <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}"> Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>⋅</mo> <mn>2</mn> <mo>=</mo> <mn>16.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01e831cda0f109e2b9cbb70877c9d4ccf049f965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:21.073ex; height:2.676ex;" alt="{\displaystyle 2^{4}=2\cdot 2\cdot 2\cdot 2=16.}"> Тут показником степені є число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dc0881187d7c549d76cf6e7274b30039a73853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 4,}"> а основою число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f2b3373a07e65d3312989163b5ebd400af86480" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 2.}"> У випадку, якщо основа - від'ємне число, наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-3)^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-3)^{3},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4a60d7898f8ba96c5c6ee06f7b8c210dee0907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.481ex; height:3.176ex;" alt="{\displaystyle (-3)^{3},}"> то необхідно враховувати операцію обернення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04bd52ce670743d3b61bec928a7ec9f47309eb36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle -}">, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-3)\cdot (-3)\cdot (-3)=-(-(-(3\cdot 3\cdot 3)))=-27.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>27.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-3)\cdot (-3)\cdot (-3)=-(-(-(3\cdot 3\cdot 3)))=-27.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbce009ed8e6d534497adabeb92534cee1cd3a2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.373ex; height:2.843ex;" alt="{\displaystyle (-3)\cdot (-3)\cdot (-3)=-(-(-(3\cdot 3\cdot 3)))=-27.}"> Степінь числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> із показником степені <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b803da9c45c1186883bde55107e9ccb102c92c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.877ex; height:1.676ex;" alt="{\displaystyle a.}"> Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-13)^{1}=-13.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>13</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>13.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-13)^{1}=-13.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/841ceb403ca8b4f68a0ad122a86eec7cefa85436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.875ex; height:3.176ex;" alt="{\displaystyle (-13)^{1}=-13.}"> Якщо вважати <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} \cup \{0\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} \cup \{0\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14021068fdcf404ee5fef81bce728414909549af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.63ex; height:2.843ex;" alt="{\displaystyle n\in \mathbb {N} \cup \{0\},}"> то існує правило, що якщо показник степеня із основою <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5920e98ff3dd1cb41e01f76243300450c958d5e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.676ex;" alt="{\displaystyle n\neq 0}"> дорівнює нулю, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{0},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1eab804165735d07766e1e65d03ac40790892ef1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.096ex; height:3.009ex;" alt="{\displaystyle n^{0},}"> то <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n^{0}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{0}=1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cf6597cddf49ae0f6481a517028625a67a5921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.357ex; height:2.676ex;" alt="{\displaystyle n^{0}=1.}"> Для будь-якого цілого <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та натуральних чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> справджується рівність <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}\cdot a^{k}=a^{n+k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}\cdot a^{k}=a^{n+k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a78d9c53f801ff7fda5da128976716003e6964ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.775ex; height:2.676ex;" alt="{\displaystyle a^{n}\cdot a^{k}=a^{n+k}.}"> Зокрема, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\cdot \underbrace {a\cdot a\cdot \cdot \cdot a} _{k\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n+k\,{\text{разів}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>⋅</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\cdot \underbrace {a\cdot a\cdot \cdot \cdot a} _{k\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n+k\,{\text{разів}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88a604b134209acde27ece85e4a0ac387ecb9afb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:33.545ex; height:6.009ex;" alt="{\displaystyle \underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\cdot \underbrace {a\cdot a\cdot \cdot \cdot a} _{k\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n+k\,{\text{разів}}}.}"> Така властивість степеня називається основною і формулюється наступним чином: щоб помножити степені з однаковими основами, необхідно основу залишити, а показники степенів додати. Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}\cdot 3^{1}=3^{2+1}=3^{3}=27,\quad \quad 2^{2}\cdot 2^{3}=2^{2+3}=2^{5}=32,\quad b^{3}\cdot b^{25}=b^{3+25}=b^{28},\quad \quad 2^{2}\cdot 2^{3}\cdot 2^{4}=2^{2+3+4}=2^{9}=512.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>27</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mn>32</mn> <mo>,</mo> <mspace width="1em" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>25</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mn>25</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>28</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>=</mo> <mn>512.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}\cdot 3^{1}=3^{2+1}=3^{3}=27,\quad \quad 2^{2}\cdot 2^{3}=2^{2+3}=2^{5}=32,\quad b^{3}\cdot b^{25}=b^{3+25}=b^{28},\quad \quad 2^{2}\cdot 2^{3}\cdot 2^{4}=2^{2+3+4}=2^{9}=512.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a6c2670e98803f9d6ab0bc4125dc7373212e6f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:115.97ex; height:3.009ex;" alt="{\displaystyle 3^{2}\cdot 3^{1}=3^{2+1}=3^{3}=27,\quad \quad 2^{2}\cdot 2^{3}=2^{2+3}=2^{5}=32,\quad b^{3}\cdot b^{25}=b^{3+25}=b^{28},\quad \quad 2^{2}\cdot 2^{3}\cdot 2^{4}=2^{2+3+4}=2^{9}=512.}"> Для будь-якого цілого <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f455a7f96d74aa94573d8e32da3b240ab0aa294f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.491ex; height:2.676ex;" alt="{\displaystyle a\neq 0}"> і натуральних <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e185ab9c990830d5055fa3ae698a4225ce67e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.858ex; height:2.509ex;" alt="{\displaystyle k,}"> за умови <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>k,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mi>k</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>k,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4b6c3140e6abfa5f36501d3f33ce833c12ef33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.351ex; height:2.509ex;" alt="{\displaystyle n>k,}"> справедлива рівність <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n}:a^{k}=a^{n-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}:a^{k}=a^{n-k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91ecc47dace635bcee6c34b58d023ebed41adc80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.033ex; height:2.676ex;" alt="{\displaystyle a^{n}:a^{k}=a^{n-k}.}"> Зокрема, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{n-k}\cdot a^{k}=a^{n-k+k}=a^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n-k}\cdot a^{k}=a^{n-k+k}=a^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/835ffd65bfcec10343dcb24d6888adea6b16c65d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.591ex; height:2.676ex;" alt="{\displaystyle a^{n-k}\cdot a^{k}=a^{n-k+k}=a^{n}.}"> Таким чином, щоб розділити степені з однаковими основами, необхідно основу залишити, а від показника степеня діленого відняти показник степеня дільника. Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5^{4}:5^{2}=5^{2}=25,\quad \quad s^{32}:s^{14}=s^{32-14}=s^{18}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>:</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>25</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> <mo>:</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>14</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> <mo>−</mo> <mn>14</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5^{4}:5^{2}=5^{2}=25,\quad \quad s^{32}:s^{14}=s^{32-14}=s^{18}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6797f793da654e9cfafcb6f095b4731827d5ed45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.358ex; height:3.009ex;" alt="{\displaystyle 5^{4}:5^{2}=5^{2}=25,\quad \quad s^{32}:s^{14}=s^{32-14}=s^{18}.}"> Щоб піднести степінь до степеня, необхідно основу залишити, а показники степенів перемножити: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{n})^{k}=a^{n\cdot k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>⋅</mo> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{n})^{k}=a^{n\cdot k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37f33ce92f1630fab9746a0620b2daa366c8532c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.854ex; height:3.176ex;" alt="{\displaystyle (a^{n})^{k}=a^{n\cdot k}.}"> Зокрема, наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{3})^{8}=x^{3\cdot 8}=x^{24},\quad \quad (2^{2})^{3}=2^{2\cdot 3}=2^{6}=64,\quad \quad (b^{3n+1})^{6}=b^{6(3n+1)}=b^{18n+6}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>⋅</mo> <mn>8</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <mn>64</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>18</mn> <mi>n</mi> <mo>+</mo> <mn>6</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{3})^{8}=x^{3\cdot 8}=x^{24},\quad \quad (2^{2})^{3}=2^{2\cdot 3}=2^{6}=64,\quad \quad (b^{3n+1})^{6}=b^{6(3n+1)}=b^{18n+6}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eb614cd20248d80571fed474d4b3ecf736bc45e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:80.131ex; height:3.343ex;" alt="{\displaystyle (x^{3})^{8}=x^{3\cdot 8}=x^{24},\quad \quad (2^{2})^{3}=2^{2\cdot 3}=2^{6}=64,\quad \quad (b^{3n+1})^{6}=b^{6(3n+1)}=b^{18n+6}.}"> Таким чином, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{n})^{k}=\underbrace {a^{n}\cdot a^{n}\cdot \cdot \cdot a^{n}} _{k\,{\text{разів}}}=a^{\overbrace {n+n+...+n} ^{k\,{\text{разів}}}}=a^{nk}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>n</mi> <mo>+</mo> <mi>n</mi> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mi>n</mi> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </mover> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>k</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{n})^{k}=\underbrace {a^{n}\cdot a^{n}\cdot \cdot \cdot a^{n}} _{k\,{\text{разів}}}=a^{\overbrace {n+n+...+n} ^{k\,{\text{разів}}}}=a^{nk}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d98da0a3baf4c3c30d51cbd7ec43b1bfa5ea3ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:41.251ex; height:10.176ex;" alt="{\displaystyle (a^{n})^{k}=\underbrace {a^{n}\cdot a^{n}\cdot \cdot \cdot a^{n}} _{k\,{\text{разів}}}=a^{\overbrace {n+n+...+n} ^{k\,{\text{разів}}}}=a^{nk}.}"> Операція піднесення до степеня є дією третього ступеня і має більший пріорітет, ніж множення та ділення. Таким чином, додавання та віднімання - дії першого ступеня, множення та ділення - дії другого ступеня, а піднесення до степеня - дія третього ступеня. Для будь-яких цілих чисел <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> та натурального числа <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> справедлива рівність: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ab)^{n}=a^{n}b^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)^{n}=a^{n}b^{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2bf22a3d4d92911ef409ba0b0353565b25c3bad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.665ex; height:2.843ex;" alt="{\displaystyle (ab)^{n}=a^{n}b^{n}.}"> Таким чином, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (ab)^{n}=\underbrace {ab\cdot ab\cdot \cdot \cdot ab} _{n\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\,\cdot \,\underbrace {b\cdot b\cdot \cdot \cdot b} _{n\,{\text{разів}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mi>b</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> <mi>b</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>a</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>b</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mi>b</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>разів</mtext> </mrow> </mrow> </munder> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ab)^{n}=\underbrace {ab\cdot ab\cdot \cdot \cdot ab} _{n\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\,\cdot \,\underbrace {b\cdot b\cdot \cdot \cdot b} _{n\,{\text{разів}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52abc5e7a3e3163964c3361fff68b17c17a555fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:44.969ex; height:6.676ex;" alt="{\displaystyle (ab)^{n}=\underbrace {ab\cdot ab\cdot \cdot \cdot ab} _{n\,{\text{разів}}}=\underbrace {a\cdot a\cdot \cdot \cdot a} _{n\,{\text{разів}}}\,\cdot \,\underbrace {b\cdot b\cdot \cdot \cdot b} _{n\,{\text{разів}}}.}"> Щоб піднести до степеня добуток, необхідно піднести до цього степеня кожний множник та перемножити їх. Зокрема, наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3xy)^{4}=3^{4}x^{4}y^{4}=81x^{4}y^{4},\quad \quad (i^{2}j^{0}kl)^{17}=i^{\,2\cdot 17}j^{\,0\cdot 17}k^{17}l^{17}=i^{34}1k^{17}l^{17}=i^{34}k^{17}l^{17}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>81</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo stretchy="false">(</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mi>k</mi> <mi>l</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>2</mn> <mo>⋅</mo> <mn>17</mn> </mrow> </msup> <msup> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mspace width="thinmathspace" /> <mn>0</mn> <mo>⋅</mo> <mn>17</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msup> <mn>1</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <msup> <mi>l</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>17</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3xy)^{4}=3^{4}x^{4}y^{4}=81x^{4}y^{4},\quad \quad (i^{2}j^{0}kl)^{17}=i^{\,2\cdot 17}j^{\,0\cdot 17}k^{17}l^{17}=i^{34}1k^{17}l^{17}=i^{34}k^{17}l^{17}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/211afd48bd040e281038506faf9865423d7ca325" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:83.864ex; height:3.176ex;" alt="{\displaystyle (3xy)^{4}=3^{4}x^{4}y^{4}=81x^{4}y^{4},\quad \quad (i^{2}j^{0}kl)^{17}=i^{\,2\cdot 17}j^{\,0\cdot 17}k^{17}l^{17}=i^{34}1k^{17}l^{17}=i^{34}k^{17}l^{17}.}"> <div class="mw-heading mw-heading2"><h2 id="Мономи_(одночлени)">Мономи (одночлени)</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=6" title="Редагувати розділ: Мономи (одночлени)">ред.</a>]</div> Мономом називається добуток чисел, змінних та їхніх степенів. Окреме число або змінна також є мономом. Наприклад, розгляньмо дві групи виразів: першу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}bc,\quad \quad 21,\quad \quad 4b^{2},\quad \quad -kl2m,\quad \quad t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>21</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mo>−</mo> <mi>k</mi> <mi>l</mi> <mn>2</mn> <mi>m</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}bc,\quad \quad 21,\quad \quad 4b^{2},\quad \quad -kl2m,\quad \quad t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/975c73da63266c194dee9032af26cf3f83c0af94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.299ex; height:3.009ex;" alt="{\displaystyle a^{2}bc,\quad \quad 21,\quad \quad 4b^{2},\quad \quad -kl2m,\quad \quad t}"> та другу <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15+d,\quad \quad {\frac {b}{6}},\quad \quad a:4,\quad \quad 2ab-xy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>+</mo> <mi>d</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mn>6</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mi>a</mi> <mo>:</mo> <mn>4</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15+d,\quad \quad {\frac {b}{6}},\quad \quad a:4,\quad \quad 2ab-xy.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41349fbcf491d27df35ce016806402a81879c602" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.109ex; height:5.343ex;" alt="{\displaystyle 15+d,\quad \quad {\frac {b}{6}},\quad \quad a:4,\quad \quad 2ab-xy.}"> Вирази першої групи є мономами, а вирази другої - ні, оскільки до виразів другої групи входить не добуток, а частка чи сума. У мономі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7axby}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mi>a</mi> <mi>x</mi> <mi>b</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7axby}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad56a08afd768ec49071fe92007056c03943aa75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.875ex; height:2.509ex;" alt="{\displaystyle 7axby}"> числовий множник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1341efd8b44bd78556844e2ad48200b6da40db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 7,}"> записаний першим, називається коефіцієнтом монома. Коефіцієнт, записаний першим, разом із степенями змінних, називається мономом стандартного вигляду. Наприклад, серед мономів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3a)^{3}25abc,\quad \quad 17abcd,\quad \quad 5m^{2}n\cdot (-3)m^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>25</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>17</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>5</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3a)^{3}25abc,\quad \quad 17abcd,\quad \quad 5m^{2}n\cdot (-3)m^{4}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee23401ccb71c24f30577fd85b7ecbdf2a73ef71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.154ex; height:3.176ex;" alt="{\displaystyle (3a)^{3}25abc,\quad \quad 17abcd,\quad \quad 5m^{2}n\cdot (-3)m^{4}}"> мономом стандартного вигляду є моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 17abcd.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>17</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 17abcd.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c943223bd1a553108cf827dead6c7b135edeb4c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.422ex; height:2.176ex;" alt="{\displaystyle 17abcd.}"> Тобто мономом стандартного вигляду називається такий моном, який містить лише один числовий множник, розміщений на першому місці ліворуч, і степені різних змінних. Щоб подати моном у стандартному вигляді, необхідно знайти його коефіцієнт, який дорівнює добутку усіх числових множників, знайти добутки всіх степенів однакових змінних й записати результат у вигляді добутку, перший множник якого - отриманий коефіцієнт. Наприклад, подамо моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3a)^{3}25abc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mn>25</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3a)^{3}25abc}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/890c43dffed5f13a43b7c5dd3abd86d759a7b1e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.815ex; height:3.176ex;" alt="{\displaystyle (3a)^{3}25abc}"> у стандартному вигляді. Для початку зведімо у степінь <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (3a)^{3}=27a^{3}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>27</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3a)^{3}=27a^{3}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f25fd37321823a4e0ec6f2ab866ffa35bd1b3fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.61ex; height:3.176ex;" alt="{\displaystyle (3a)^{3}=27a^{3}.}"> Згрупуймо подібні множники таким чином, щоб числовий множник знаходився ліворуч: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 27\cdot 25\cdot a^{3}\cdot a\cdot b\cdot c=675a^{4}bc.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>27</mn> <mo>⋅</mo> <mn>25</mn> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>⋅</mo> <mi>c</mi> <mo>=</mo> <mn>675</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>b</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 27\cdot 25\cdot a^{3}\cdot a\cdot b\cdot c=675a^{4}bc.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/372da83c0dbd0c5e3bb462745d4d7a22eba195d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:30.085ex; height:2.676ex;" alt="{\displaystyle 27\cdot 25\cdot a^{3}\cdot a\cdot b\cdot c=675a^{4}bc.}"> Таким чином, моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 675a^{4}bc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>675</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 675a^{4}bc}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7341ac47a67d833514952b5fa97c5b50aef6419a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.776ex; height:2.676ex;" alt="{\displaystyle 675a^{4}bc}"> є мономом стандартного вигляду. Зробімо те саме із мономом <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5m^{2}n\cdot (-3)m^{4}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5m^{2}n\cdot (-3)m^{4}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51cd33762da10342bd583d035635b6e616c439e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.852ex; height:3.176ex;" alt="{\displaystyle 5m^{2}n\cdot (-3)m^{4}.}"> Отримаємо <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\cdot (-3)\cdot m^{2}\cdot m^{4}\cdot n=-15m^{6}n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>⋅</mo> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mn>15</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\cdot (-3)\cdot m^{2}\cdot m^{4}\cdot n=-15m^{6}n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7f34893821cc86f82a13f450f47750256095fc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.61ex; height:3.176ex;" alt="{\displaystyle 5\cdot (-3)\cdot m^{2}\cdot m^{4}\cdot n=-15m^{6}n.}"> Моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -15m^{6}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>15</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -15m^{6}n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ffb837f71fcc91a28aeac3d17e3923f3d4e0359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.622ex; height:2.843ex;" alt="{\displaystyle -15m^{6}n}"> є мономом стандартного вигляду. Степенем монома є сума показників степенів змінних, що входять до його складу. Якщо мономом є число, відмінне від нуля, то степінь монома вважають рівним нулю. Наприклад, у мономі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16a^{2}bc^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16a^{2}bc^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24c7a155b846a4aadeeccab85e69466d24aba3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.668ex; height:2.676ex;" alt="{\displaystyle 16a^{2}bc^{3}}"> є три змінні - <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/181523deba732fda302fd176275a0739121d3bc8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.261ex; height:2.509ex;" alt="{\displaystyle a,b}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"> Показник степеня <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/993d53082bc05c266933af9a892e1ce2128547cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 2,}"> показник степеня <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc5fd8163a83100c5330622e9e317fa4e872403" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 1,}"> показник степеня <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75e17f7e7094c457fa7c8f743c3d8622eebd8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 3.}"> Відтак, степінь монома буде дорівнювати: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2+1+3=6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2+1+3=6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59f5f7e770145e213bfd87bfa062a668ce9412f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.076ex; height:2.343ex;" alt="{\displaystyle 2+1+3=6.}"> Виконуючи дії із мономами, необхідно використовувати властивості дій множення та піднесення до степеня, оскільки моном складається лише з таких дій. Моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16a^{2}bc^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16a^{2}bc^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b24c7a155b846a4aadeeccab85e69466d24aba3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.668ex; height:2.676ex;" alt="{\displaystyle 16a^{2}bc^{3}}"> можна записати у вигляді добутку двох мономів стандартного вигляду, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16a^{2}bc^{3}=4abc\cdot 4ac^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>⋅</mo> <mn>4</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16a^{2}bc^{3}=4abc\cdot 4ac^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68c1bf26309c3e917ca38148bc068629d2aa1689" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.942ex; height:3.009ex;" alt="{\displaystyle 16a^{2}bc^{3}=4abc\cdot 4ac^{2},}"> та у вигляді добутку трьох мономів стандартного вигляду, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 16a^{2}bc^{3}=4abc\cdot 2ac\cdot 2c.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>⋅</mo> <mn>2</mn> <mi>a</mi> <mi>c</mi> <mo>⋅</mo> <mn>2</mn> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 16a^{2}bc^{3}=4abc\cdot 2ac\cdot 2c.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be538d73c5ee41b99a85018f3609ad3f32377e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.736ex; height:2.676ex;" alt="{\displaystyle 16a^{2}bc^{3}=4abc\cdot 2ac\cdot 2c.}"> Розгляньмо декілька прикладів. Подамо наступні мономи у стандартному вигляді. <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7((x^{3})^{4})^{2}\cdot (-2(x^{4})^{3})^{2}=7(x^{3})^{8}\cdot 4(x^{4})^{6}=28x^{48};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>7</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>⋅</mo> <mn>4</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <mn>28</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>48</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7((x^{3})^{4})^{2}\cdot (-2(x^{4})^{3})^{2}=7(x^{3})^{8}\cdot 4(x^{4})^{6}=28x^{48};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce346ef8fc19df7886938f7f43e501c35d37b5a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.908ex; height:3.176ex;" alt="{\displaystyle 7((x^{3})^{4})^{2}\cdot (-2(x^{4})^{3})^{2}=7(x^{3})^{8}\cdot 4(x^{4})^{6}=28x^{48};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x^{4})^{n})^{6}\cdot 2((x^{2})^{n+2})^{5}=2(x^{4})^{6n}\cdot (x^{2})^{5n+10}=2x^{24n}\cdot x^{10n+20}=2x^{24n+10n+20}=2x^{34n+20};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>⋅</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>10</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>20</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>24</mn> <mi>n</mi> <mo>+</mo> <mn>10</mn> <mi>n</mi> <mo>+</mo> <mn>20</mn> </mrow> </msup> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>34</mn> <mi>n</mi> <mo>+</mo> <mn>20</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((x^{4})^{n})^{6}\cdot 2((x^{2})^{n+2})^{5}=2(x^{4})^{6n}\cdot (x^{2})^{5n+10}=2x^{24n}\cdot x^{10n+20}=2x^{24n+10n+20}=2x^{34n+20};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ccfb61b7e13e6b84500637b1c0c17265305499" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:87.376ex; height:3.176ex;" alt="{\displaystyle ((x^{4})^{n})^{6}\cdot 2((x^{2})^{n+2})^{5}=2(x^{4})^{6n}\cdot (x^{2})^{5n+10}=2x^{24n}\cdot x^{10n+20}=2x^{24n+10n+20}=2x^{34n+20};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((x^{2n+3})^{4})^{n}=(x^{2n+3})^{4n}=x^{4n(2n+3)}=x^{8n^{2}+12n};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>n</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>n</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ((x^{2n+3})^{4})^{n}=(x^{2n+3})^{4n}=x^{4n(2n+3)}=x^{8n^{2}+12n};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a1d7d4f5f8873e6c9ea95e707c93340b28cbaea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.293ex; height:3.509ex;" alt="{\displaystyle ((x^{2n+3})^{4})^{n}=(x^{2n+3})^{4n}=x^{4n(2n+3)}=x^{8n^{2}+12n};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a^{n+3}b^{2n})^{3}\cdot 4a^{3n}\cdot (-2b)^{2}=16a^{3n+9}\cdot a^{3n}b^{6n}\cdot b^{2}=16a^{3n+3n+9}b^{6n+2}=16a^{6n+9}b^{6n+2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> </mrow> </msup> <mo>⋅</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>3</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> <mo>+</mo> <mn>9</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a^{n+3}b^{2n})^{3}\cdot 4a^{3n}\cdot (-2b)^{2}=16a^{3n+9}\cdot a^{3n}b^{6n}\cdot b^{2}=16a^{3n+3n+9}b^{6n+2}=16a^{6n+9}b^{6n+2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e667f2f6e15d3e51e0972f05115d1e1c909b27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:83.184ex; height:3.176ex;" alt="{\displaystyle (a^{n+3}b^{2n})^{3}\cdot 4a^{3n}\cdot (-2b)^{2}=16a^{3n+9}\cdot a^{3n}b^{6n}\cdot b^{2}=16a^{3n+3n+9}b^{6n+2}=16a^{6n+9}b^{6n+2}.}"></li></ul> <div class="mw-heading mw-heading2"><h2 id="Поліноми_(многочлени)">Поліноми (многочлени)</h2>[<a href="/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&action=edit&section=7" title="Редагувати розділ: Поліноми (многочлени)">ред.</a>]</div> Поліномом називається сума кількох мономів. Наприклад, вираз <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+4ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+4ac}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4070f6f49b458827faab58e2a829221a319b81c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.623ex; height:2.843ex;" alt="{\displaystyle x^{4}+4ac}"> складається із суми двох мономів: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf0bf28fd28f45d07e1ceb909ce333c18c558c93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.676ex;" alt="{\displaystyle x^{2}}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4ac,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>a</mi> <mi>c</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4ac,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20d60c9c9e0016e3907697c92dc50be1282b272" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.046ex; height:2.509ex;" alt="{\displaystyle 4ac,}"> які називаються членами даного полінома. Членами полінома <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{3}-3ab-4a+10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{3}-3ab-4a+10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/464b9b1b56b7fb08ba86b3add3184eea8b07af63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.075ex; height:2.843ex;" alt="{\displaystyle 2a^{3}-3ab-4a+10}"> є <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{3},\,\,-3ab,\,\,-4a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mo>−</mo> <mn>4</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{3},\,\,-3ab,\,\,-4a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e195557c0a4c83610364ff3dd82ad751e26253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.461ex; height:3.009ex;" alt="{\displaystyle 2a^{3},\,\,-3ab,\,\,-4a}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6314920d56a18621f0fbc1d56526be049949f2b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.176ex;" alt="{\displaystyle 10.}"> Вважається, що поліном може складатися лише з одного монома та що кожний моном є поліномом, який складається з одного члена. Розгляньмо поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10ab-4+a-4ab+3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>a</mi> <mi>b</mi> <mo>−</mo> <mn>4</mn> <mo>+</mo> <mi>a</mi> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10ab-4+a-4ab+3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa1efb3ceba42313693067029829528e90f5de7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.505ex; height:2.343ex;" alt="{\displaystyle 10ab-4+a-4ab+3.}"> У цьому мономі є подібні доданки: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156827a30e9f6cbe167c05c6f8528ca430cb72a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.552ex; height:2.176ex;" alt="{\displaystyle 10ab}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4ab,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4ab,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1a46090d7bd32650c5890a7d8ab249fed125db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.845ex; height:2.509ex;" alt="{\displaystyle -4ab,}"> а також <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75e17f7e7094c457fa7c8f743c3d8622eebd8ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 3.}"> Доданки <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/156827a30e9f6cbe167c05c6f8528ca430cb72a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.552ex; height:2.176ex;" alt="{\displaystyle 10ab}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/920b4d550b2be823a4d26af45c9b5dca470bee9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.198ex; height:2.343ex;" alt="{\displaystyle -4ab}"> відрізняються числовими множниками (коефіцієнтами мономів, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec811eb07dcac7ea67b413c5665390a1671ecb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 10}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/addc8519724f81b43a6883c5eb2c996f9fc2996f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle -4}">). Подібні доданки у поліномі називаються подібними членами полінома. Поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-4ax^{2}+b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-4ax^{2}+b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f55f6b8e999e56d8f5077036260a54f0b356bbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.838ex; height:2.843ex;" alt="{\displaystyle x^{2}-4ax^{2}+b}"> вже не містить подібних доданків. Однак обидва наведених поліноми складаються із мономів стандартного вигляду. Степем полінома є найбільший із степенів мономів, які є членами даного полінома. Наприклад, поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{2}b^{2}+b^{3}-2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{2}b^{2}+b^{3}-2a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7899c4888c08c28419f6f664709a80c1527eacd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:2.843ex;" alt="{\displaystyle 2a^{2}b^{2}+b^{3}-2a}"> складається з тьох мономів: першого <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{2}b^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{2}b^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d11c0ad2cbc96ae9a444daae7eee5d1a3a06b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.145ex; height:3.009ex;" alt="{\displaystyle 2a^{2}b^{2},}"> другого <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0857fac5684e0a05b0d6cadea4b4ccf170edd4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.052ex; height:2.676ex;" alt="{\displaystyle b^{3}}"> й третього <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -2a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -2a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6128fb6713604d0d5ffef7ab39a4b445ff92f1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.847ex; height:2.343ex;" alt="{\displaystyle -2a.}"> Степінь першого монома дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dc0881187d7c549d76cf6e7274b30039a73853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 4,}"> степінь другого монома дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e86384bc2ac4ac6c2a68904ba067110f0876bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 3,}"> степінь третього монома дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af8c4e445819b13a052647aa3eb2be990b0a4b24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 1.}"> Найбільшим за степенем мономом є моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{2}b^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{2}b^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d11c0ad2cbc96ae9a444daae7eee5d1a3a06b96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.145ex; height:3.009ex;" alt="{\displaystyle 2a^{2}b^{2},}"> степінь якого дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cad196d766a835649677e286e4ea30bdaa99d3e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 4.}"> Відтак й степінь полінома <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{2}b^{2}+b^{3}-2a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{2}b^{2}+b^{3}-2a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7899c4888c08c28419f6f664709a80c1527eacd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.623ex; height:2.843ex;" alt="{\displaystyle 2a^{2}b^{2}+b^{3}-2a}"> дорівнює <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6dc0881187d7c549d76cf6e7274b30039a73853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.809ex; height:2.509ex;" alt="{\displaystyle 4,}"> тобто дорівнює найбільшому із степенів мономів, які є його членами. Поліном стандартного вигляду - це сума мономів, що утворюють даний поліном, серед яких немає подібних членів і які розташовані у порядку спадання чи зростання показників степенів членів. Наприклад, серед поліномів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7x^{4}+x^{3}-10x^{2}+x-5,\quad \quad 15+2a-4ac^{4}+c^{3}-a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mspace width="1em" /> <mspace width="1em" /> <mn>15</mn> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>4</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7x^{4}+x^{3}-10x^{2}+x-5,\quad \quad 15+2a-4ac^{4}+c^{3}-a^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa2201a1e7f1f436bdb91649f8f6e68259da14b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:55.049ex; height:3.009ex;" alt="{\displaystyle 7x^{4}+x^{3}-10x^{2}+x-5,\quad \quad 15+2a-4ac^{4}+c^{3}-a^{2}}"> поліномом стандартного вигляду є поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 7x^{4}+x^{3}-10x^{2}+x+5.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>5.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 7x^{4}+x^{3}-10x^{2}+x+5.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a70ce8280b7fa28c05257619e81e7a7df5ab5f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:25.14ex; height:2.843ex;" alt="{\displaystyle 7x^{4}+x^{3}-10x^{2}+x+5.}"> Поліном ж <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15+2a-5ac^{4}+c^{3}-a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>5</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15+2a-5ac^{4}+c^{3}-a^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecf2356012b575c30b127845271ab1a8799a2df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:24.877ex; height:2.843ex;" alt="{\displaystyle 15+2a-5ac^{4}+c^{3}-a^{2}}"> не є поліномом стандартного вигляду, оскільки мономи стандартного вигляду, з яких він складається, не розташовані у порядку зростання (спадання) показників степенів. Кожний поліном є цілим виразом, але не кожний цілий вираз може називатися поліномом. Наприклад, цілі вирази <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+1)(x-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+1)(x-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5abc7ede078c30f6f2bbb39bf29b745c33849e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.284ex; height:2.843ex;" alt="{\displaystyle (x+1)(x-1)}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-b)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-b)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd517db26f9b9b3013d12b69c31b4a4699d9fd85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.031ex; height:3.176ex;" alt="{\displaystyle (x-b)^{2}}"> не є поліномами, оскільки вони не є сумами мономів. Щоб додати поліноми, необхідно звести подібні доданки, які є членами даних поліномів. Наприклад, додамо поліноми <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12a^{2}-5a+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12a^{2}-5a+7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b40d9dba6e8bbe987fd3588f07fa8f748f9ed46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.844ex; height:2.843ex;" alt="{\displaystyle 12a^{2}-5a+7}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4a^{2}+2a+20:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4a^{2}+2a+20:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e249696d5de713ebbb3fb71e9e3ca9bdfec0c78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.945ex; height:2.843ex;" alt="{\displaystyle -4a^{2}+2a+20:}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (12a^{2}-5a+7)+(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{-}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{+}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{+}\color {green}{20}=\color {red}{8a^{2}}\color {black}{-}\color {blue}{3a}\color {black}{+}\color {green}{27}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> <mo>=</mo> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>27</mn> </mrow> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (12a^{2}-5a+7)+(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{-}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{+}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{+}\color {green}{20}=\color {red}{8a^{2}}\color {black}{-}\color {blue}{3a}\color {black}{+}\color {green}{27}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54aef344067f49eddb6c2c8c3e17315cad642e64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:79.948ex; height:3.176ex;" alt="{\displaystyle (12a^{2}-5a+7)+(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{-}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{+}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{+}\color {green}{20}=\color {red}{8a^{2}}\color {black}{-}\color {blue}{3a}\color {black}{+}\color {green}{27}.}"> За віднімання здійснюється така сама процедура зведення подібних членів полінома. Наприклад, віднімемо від полінома <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12a^{2}-5a+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12a^{2}-5a+7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b40d9dba6e8bbe987fd3588f07fa8f748f9ed46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.844ex; height:2.843ex;" alt="{\displaystyle 12a^{2}-5a+7}"> поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4a^{2}+2a+20:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4a^{2}+2a+20:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e249696d5de713ebbb3fb71e9e3ca9bdfec0c78e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.945ex; height:2.843ex;" alt="{\displaystyle -4a^{2}+2a+20:}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (12a^{2}-5a+7)-(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{+}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{-}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{-}\color {green}{20}=\color {red}{16a^{2}}\color {black}{-}\color {blue}{7a}\color {black}{-}\color {green}{13}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>20</mn> </mrow> <mo>=</mo> <mstyle mathcolor="red"> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="blue"> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> <mi>a</mi> </mrow> <mstyle mathcolor="black"> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mstyle mathcolor="green"> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mo>.</mo> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (12a^{2}-5a+7)-(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{+}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{-}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{-}\color {green}{20}=\color {red}{16a^{2}}\color {black}{-}\color {blue}{7a}\color {black}{-}\color {green}{13}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1141701e66d5f65ae9e278c2626542f7901d7b06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:81.11ex; height:3.176ex;" alt="{\displaystyle (12a^{2}-5a+7)-(-4a^{2}+2a+20)=\color {red}{12a^{2}}\color {black}{+}\color {red}{4a^{2}}\color {black}{-}\color {blue}{5a}\color {black}{-}\color {blue}{2a}\color {black}{+}\color {green}{7}\color {black}{-}\color {green}{20}=\color {red}{16a^{2}}\color {black}{-}\color {blue}{7a}\color {black}{-}\color {green}{13}.}"> Зверніть увагу, що мінус перед дужками обертає доданки, які в них містяться, на протилежні. Тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -(-4a^{2}+2a+20)=4a^{2}-2a-20.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>20.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(-4a^{2}+2a+20)=4a^{2}-2a-20.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f37459754fb195a56d8179e704d6190c3d377381" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.86ex; height:3.176ex;" alt="{\displaystyle -(-4a^{2}+2a+20)=4a^{2}-2a-20.}"> Зрозуміло, що додати до полінома <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 12a^{2}-5a+7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 12a^{2}-5a+7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b40d9dba6e8bbe987fd3588f07fa8f748f9ed46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.844ex; height:2.843ex;" alt="{\displaystyle 12a^{2}-5a+7}"> поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4a^{2}+2a+20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4a^{2}+2a+20}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7abcf6e9b72a292686315241c9378063a8c57f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.653ex; height:2.843ex;" alt="{\displaystyle -4a^{2}+2a+20}"> - означає відняти від нього поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4a^{2}-2a-20,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>20</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4a^{2}-2a-20,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c41827fecbce713ea644878993c3c0e56a6ff92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.491ex; height:3.009ex;" alt="{\displaystyle 4a^{2}-2a-20,}"> тобто наступна рівність є тотожністю: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (12a^{2}-5a+7)-(4a^{2}-2a-20)=(12a^{2}-5a+7)+(-4a+2a+20).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>−</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>12</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>7</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>20</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (12a^{2}-5a+7)-(4a^{2}-2a-20)=(12a^{2}-5a+7)+(-4a+2a+20).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d9cb4f31125c7211a6194f3dcb58ebb8f3bd809" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.795ex; height:3.176ex;" alt="{\displaystyle (12a^{2}-5a+7)-(4a^{2}-2a-20)=(12a^{2}-5a+7)+(-4a+2a+20).}"> Щоб помножити моном на поліном, потрібно помножити його на кожний член полінома і отримані добутки додати. Наприклад, помножимо моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6x^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350f29756e1193125113bc7a9d59997023e06292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" alt="{\displaystyle 6x^{2}}"> на поліном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{4}+8axy-7ac.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>7</mn> <mi>a</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{4}+8axy-7ac.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d85965663252d9eb0c2ec30729c1a1812d39cde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.819ex; height:3.009ex;" alt="{\displaystyle y^{4}+8axy-7ac.}"> Використаймо розподільну властивість множення, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(b+c)=ab+ac.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(b+c)=ab+ac.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671d624300064920a8d7a1ae9cc11b397f4b83ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.934ex; height:2.843ex;" alt="{\displaystyle a(b+c)=ab+ac.}"> Одержимо: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{2}(y^{4}+8axy-7ac)=6x^{2}\cdot y^{4}+6x^{2}\cdot 8axy-6x^{2}\cdot 7ac=6x^{2}y^{4}+48ax^{3}y-42acx^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>7</mn> <mi>a</mi> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>8</mn> <mi>a</mi> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mn>7</mn> <mi>a</mi> <mi>c</mi> <mo>=</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>48</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>42</mn> <mi>a</mi> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6x^{2}(y^{4}+8axy-7ac)=6x^{2}\cdot y^{4}+6x^{2}\cdot 8axy-6x^{2}\cdot 7ac=6x^{2}y^{4}+48ax^{3}y-42acx^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ac76bbd3d69642f381a7cecb88f7a11e80d731" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.702ex; height:3.176ex;" alt="{\displaystyle 6x^{2}(y^{4}+8axy-7ac)=6x^{2}\cdot y^{4}+6x^{2}\cdot 8axy-6x^{2}\cdot 7ac=6x^{2}y^{4}+48ax^{3}y-42acx^{2}.}"> Взагалі, множення поліномів є комутативним, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b=b\cdot a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=b\cdot a.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9a7a9f915468093b88dd498329a98aaa386fcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.558ex; height:2.176ex;" alt="{\displaystyle a\cdot b=b\cdot a.}"> Помножимо два поліноми <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8u^{5}v+16uv^{4}+v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>v</mi> <mo>+</mo> <mn>16</mn> <mi>u</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8u^{5}v+16uv^{4}+v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9faf09fe461c757f6df0def1328b82642d4ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.319ex; height:2.843ex;" alt="{\displaystyle 8u^{5}v+16uv^{4}+v}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9b^{7}u^{3}:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9b^{7}u^{3}:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41417784563621c58bbb578ae1f6eccf59b15b63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.89ex; height:2.676ex;" alt="{\displaystyle 9b^{7}u^{3}:}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (8u^{5}v+16uv^{4}+v)\cdot 9b^{7}u^{3}=9b^{7}u^{3}\cdot (8u^{5}v+16uv^{4}+v)=9b^{7}u^{3}\cdot 8u^{5}v+9b^{7}u^{3}\cdot 16uv^{4}+9b^{7}u^{3}\cdot v=72b^{7}u^{8}v+144b^{7}u^{4}v^{4}+9b^{7}u^{3}v.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>8</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>v</mi> <mo>+</mo> <mn>16</mn> <mi>u</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>8</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>v</mi> <mo>+</mo> <mn>16</mn> <mi>u</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>8</mn> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mi>v</mi> <mo>+</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>16</mn> <mi>u</mi> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mi>v</mi> <mo>=</mo> <mn>72</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mi>v</mi> <mo>+</mo> <mn>144</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (8u^{5}v+16uv^{4}+v)\cdot 9b^{7}u^{3}=9b^{7}u^{3}\cdot (8u^{5}v+16uv^{4}+v)=9b^{7}u^{3}\cdot 8u^{5}v+9b^{7}u^{3}\cdot 16uv^{4}+9b^{7}u^{3}\cdot v=72b^{7}u^{8}v+144b^{7}u^{4}v^{4}+9b^{7}u^{3}v.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b5ed2930a8941c4626f2a3c8daab78b5be054b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:132.304ex; height:3.176ex;" alt="{\displaystyle (8u^{5}v+16uv^{4}+v)\cdot 9b^{7}u^{3}=9b^{7}u^{3}\cdot (8u^{5}v+16uv^{4}+v)=9b^{7}u^{3}\cdot 8u^{5}v+9b^{7}u^{3}\cdot 16uv^{4}+9b^{7}u^{3}\cdot v=72b^{7}u^{8}v+144b^{7}u^{4}v^{4}+9b^{7}u^{3}v.}"> Щоб помножити поліном на поліном, необхідно кожний член одного полінома помножити на кожний член іншого полінома, а отримані добутки додати, звівши подібні доданки й подавши одержаний поліном у стандартному вигляді. Наприклад, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b+1)(a-b)=a\cdot a+a\cdot (-b)+b\cdot a+b\cdot (-b)+1\cdot a+1\cdot (-b)=a^{2}-ab+ab-b^{2}+a+b=a^{2}-b^{2}+a+b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>a</mi> <mo>+</mo> <mi>a</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b+1)(a-b)=a\cdot a+a\cdot (-b)+b\cdot a+b\cdot (-b)+1\cdot a+1\cdot (-b)=a^{2}-ab+ab-b^{2}+a+b=a^{2}-b^{2}+a+b.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2423eafe3f692a65f65d453b5d21cae767d63a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:116.594ex; height:3.176ex;" alt="{\displaystyle (a+b+1)(a-b)=a\cdot a+a\cdot (-b)+b\cdot a+b\cdot (-b)+1\cdot a+1\cdot (-b)=a^{2}-ab+ab-b^{2}+a+b=a^{2}-b^{2}+a+b.}"> Добуток поліномів є асоціативним. Нехай дані поліноми <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=xz^{3}+3xy^{2}-xyz,\,\,B=x^{2}z-z^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mspace width="thinmathspace" /> <mi>B</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=xz^{3}+3xy^{2}-xyz,\,\,B=x^{2}z-z^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265ee77191e656c53a1bb6dfaef0ad8fe18ee0b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.404ex; height:3.009ex;" alt="{\displaystyle A=xz^{3}+3xy^{2}-xyz,\,\,B=x^{2}z-z^{3}}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=ax+by-az.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=ax+by-az.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc4ea331e2b90734acc7fe35f7aab45f6c126f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.223ex; height:2.509ex;" alt="{\displaystyle C=ax+by-az.}"> Ми стверджуємо, що <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⋅</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43a310dc14523c1967a370b16fef7b01df4560e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.627ex; height:2.843ex;" alt="{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}"> Переконаймося у цьому. Спочатку помножимо <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot (B\cdot C):}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⋅</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot (B\cdot C):}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/912edded0546bbd79ec6be8c97ff0f65551ccf1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.733ex; height:2.843ex;" alt="{\displaystyle A\cdot (B\cdot C):}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}(xz^{3}+3xy^{2}-xyz)\cdot ((x^{2}z-z^{3})\cdot (ax+by-az))=\\=(xz^{3}+3xy^{2}-xyz)\cdot (ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{xz^{3}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}+\\\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{2}z-3ax^{3}y^{2}z^{2}-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}} _{3xy^{2}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}-\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{2}+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{-xyz(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </munder> <mo>+</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>+</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </munder> <mo>−</mo> </mtd> </mtr> <mtr> <mtd> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>.</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">(</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mi>z</mi> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}(xz^{3}+3xy^{2}-xyz)\cdot ((x^{2}z-z^{3})\cdot (ax+by-az))=\\=(xz^{3}+3xy^{2}-xyz)\cdot (ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{xz^{3}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}+\\\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{2}z-3ax^{3}y^{2}z^{2}-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}} _{3xy^{2}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}-\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{2}+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{-xyz(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c26b9e0b0283ae3d25e58b63001941887515cff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.794ex; margin-bottom: -0.211ex; width:71.054ex; height:29.176ex;" alt="{\displaystyle {\begin{matrix}(xz^{3}+3xy^{2}-xyz)\cdot ((x^{2}z-z^{3})\cdot (ax+by-az))=\\=(xz^{3}+3xy^{2}-xyz)\cdot (ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{xz^{3}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}+\\\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{2}z-3ax^{3}y^{2}z^{2}-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}} _{3xy^{2}(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}-\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{2}+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{-xyz(ax^{3}z+bx^{2}yz-ax^{2}z^{2}-axz^{3}-byz^{3}+az^{4})}\end{matrix}}}"> Помножимо тепер у порядку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A\cdot B)\cdot C:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>C</mi> <mo>:</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A\cdot B)\cdot C:}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/306317b62c3f85fb7ee98ef3459593cf786dd22e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.733ex; height:2.843ex;" alt="{\displaystyle (A\cdot B)\cdot C:}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}((xz^{3}+3xy^{2}-xyz)\cdot (x^{2}z-z^{3}))\cdot (ax+by-az)=\\=(x^{3}z^{4}-xz^{6}+3x^{3}y^{2}z-3xy^{2}z^{3}-x^{3}yz^{2}+xyz^{4})\cdot (ax+by-az)=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}} _{x^{3}z^{4}(ax+by-az)}\underbrace {-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{-xz^{6}(ax+by-az)}\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{3}z-3ax^{3}y^{2}z^{2}} _{3x^{3}y^{2}z(ax+by-az)}\underbrace {-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}-} _{-3xy^{2}z^{3}(ax+by-az)}\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{3}} _{-x^{3}yz^{2}(ax+by-az)}\underbrace {+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{xyz^{4}(ax+by-az)}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> </mtr> <mtr> <mtd> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>+</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>z</mi> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>3</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> </mtd> </mtr> <mtr> <mtd> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>.</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo>−</mo> <mi>a</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </munder> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}((xz^{3}+3xy^{2}-xyz)\cdot (x^{2}z-z^{3}))\cdot (ax+by-az)=\\=(x^{3}z^{4}-xz^{6}+3x^{3}y^{2}z-3xy^{2}z^{3}-x^{3}yz^{2}+xyz^{4})\cdot (ax+by-az)=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}} _{x^{3}z^{4}(ax+by-az)}\underbrace {-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{-xz^{6}(ax+by-az)}\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{3}z-3ax^{3}y^{2}z^{2}} _{3x^{3}y^{2}z(ax+by-az)}\underbrace {-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}-} _{-3xy^{2}z^{3}(ax+by-az)}\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{3}} _{-x^{3}yz^{2}(ax+by-az)}\underbrace {+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{xyz^{4}(ax+by-az)}\end{matrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3a97b90ead987127428615edfad56f5092ee8f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.338ex; width:117.575ex; height:21.676ex;" alt="{\displaystyle {\begin{matrix}((xz^{3}+3xy^{2}-xyz)\cdot (x^{2}z-z^{3}))\cdot (ax+by-az)=\\=(x^{3}z^{4}-xz^{6}+3x^{3}y^{2}z-3xy^{2}z^{3}-x^{3}yz^{2}+xyz^{4})\cdot (ax+by-az)=\\=\underbrace {ax^{4}z^{4}+bx^{3}yz^{4}-ax^{3}z^{5}} _{x^{3}z^{4}(ax+by-az)}\underbrace {-ax^{2}z^{6}-bxyz^{6}+axz^{7}} _{-xz^{6}(ax+by-az)}\underbrace {+3ax^{4}y^{2}z+3bx^{3}y^{3}z-3ax^{3}y^{2}z^{2}} _{3x^{3}y^{2}z(ax+by-az)}\underbrace {-3ax^{2}y^{2}z^{3}-3bxy^{3}z^{3}+3axy^{2}z^{4}-} _{-3xy^{2}z^{3}(ax+by-az)}\\\underbrace {-ax^{4}yz^{2}-bx^{3}y^{2}z^{2}+ax^{3}yz^{3}} _{-x^{3}yz^{2}(ax+by-az)}\underbrace {+ax^{2}yz^{4}+bxy^{2}z^{4}-axyz^{5}.} _{xyz^{4}(ax+by-az)}\end{matrix}}}"> В результаті виконання усіх дій ми отримали два великих поліноми, які не помістилися кожний у один ряд, тому довелося записати їх частинами із їхнім переносом у наступний рядок (зауваження: перенесення частини виразу до наступного рядку здійснюється із обов'язковим перенесенням усіх знаків). Частини виразу були підписані знизу для кращого розуміння порядку здійснення дій. Неважко побачити, що одержані поліноми у обох випадках є однаковими, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>⋅</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⋅</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>C</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d43a310dc14523c1967a370b16fef7b01df4560e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.627ex; height:2.843ex;" alt="{\displaystyle A\cdot (B\cdot C)=(A\cdot B)\cdot C.}"> Подамо отриманий поліном у стандартному вигляді, обчисливши показники степенів членів полінома й розташувавши суму мономів в порядку спадання значень їхніх показників степенів (показники степенів вказані згори над членами полінома). <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \overbrace {ax^{4}z^{4}} ^{9}\overbrace {+bx^{3}yz^{4}} ^{9}\overbrace {-ax^{3}z^{5}} ^{9}\overbrace {-ax^{2}z^{6}} ^{9}\overbrace {-bxyz^{6}} ^{9}\overbrace {+axz^{7}} ^{9}\overbrace {+3ax^{4}y^{2}z} ^{8}\overbrace {-3ax^{3}y^{2}z^{2}} ^{8}\overbrace {-3ax^{2}y^{2}z^{3}} ^{8}\overbrace {-3bxy^{3}z^{3}} ^{8}\overbrace {+3axy^{2}z^{4}} ^{8}\overbrace {-ax^{4}yz^{2}} ^{8}\overbrace {-bx^{3}y^{2}z^{2}} ^{8}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mi>a</mi> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mn>3</mn> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mn>3</mn> <mi>a</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \overbrace {ax^{4}z^{4}} ^{9}\overbrace {+bx^{3}yz^{4}} ^{9}\overbrace {-ax^{3}z^{5}} ^{9}\overbrace {-ax^{2}z^{6}} ^{9}\overbrace {-bxyz^{6}} ^{9}\overbrace {+axz^{7}} ^{9}\overbrace {+3ax^{4}y^{2}z} ^{8}\overbrace {-3ax^{3}y^{2}z^{2}} ^{8}\overbrace {-3ax^{2}y^{2}z^{3}} ^{8}\overbrace {-3bxy^{3}z^{3}} ^{8}\overbrace {+3axy^{2}z^{4}} ^{8}\overbrace {-ax^{4}yz^{2}} ^{8}\overbrace {-bx^{3}y^{2}z^{2}} ^{8}+}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9099ddb1cedc97d48bac6b22a21d927a847dd37a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:114.728ex; height:6.343ex;" alt="{\displaystyle \overbrace {ax^{4}z^{4}} ^{9}\overbrace {+bx^{3}yz^{4}} ^{9}\overbrace {-ax^{3}z^{5}} ^{9}\overbrace {-ax^{2}z^{6}} ^{9}\overbrace {-bxyz^{6}} ^{9}\overbrace {+axz^{7}} ^{9}\overbrace {+3ax^{4}y^{2}z} ^{8}\overbrace {-3ax^{3}y^{2}z^{2}} ^{8}\overbrace {-3ax^{2}y^{2}z^{3}} ^{8}\overbrace {-3bxy^{3}z^{3}} ^{8}\overbrace {+3axy^{2}z^{4}} ^{8}\overbrace {-ax^{4}yz^{2}} ^{8}\overbrace {-bx^{3}y^{2}z^{2}} ^{8}+}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \overbrace {+ax^{2}yz^{4}} ^{8}\overbrace {+bxy^{2}z^{4}} ^{8}\overbrace {-axyz^{5}} ^{8}\overbrace {+3bx^{3}y^{2}z} ^{7}\overbrace {+ax^{3}yz^{2}} ^{7}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mi>b</mi> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>−</mo> <mi>a</mi> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mn>3</mn> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mover> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mover> <mrow> <mo>+</mo> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏞</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </mover> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \overbrace {+ax^{2}yz^{4}} ^{8}\overbrace {+bxy^{2}z^{4}} ^{8}\overbrace {-axyz^{5}} ^{8}\overbrace {+3bx^{3}y^{2}z} ^{7}\overbrace {+ax^{3}yz^{2}} ^{7}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36a0c89c9e918910ab828891304710eaae83d7d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:43.908ex; height:6.343ex;" alt="{\displaystyle \overbrace {+ax^{2}yz^{4}} ^{8}\overbrace {+bxy^{2}z^{4}} ^{8}\overbrace {-axyz^{5}} ^{8}\overbrace {+3bx^{3}y^{2}z} ^{7}\overbrace {+ax^{3}yz^{2}} ^{7}.}"> Розкладання полінома на множники. Розкладання числа на множники - представлення його у вигляді добутку множників. Зокрема, ми можемо вимагати, щоб усі множники при розкладі даного числа були простими числами. Наприклад, ми можемо записати <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 45=5\cdot 3^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>45</mn> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 45=5\cdot 3^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96361ec04e52c5f3812676faca36d092fe0e75a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.128ex; height:2.676ex;" alt="{\displaystyle 45=5\cdot 3^{2}.}"> Число 45 можна записати у вигляді добутку <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15\cdot 3.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo>⋅</mo> <mn>3.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15\cdot 3.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c19f0cd5578562aca574e01856b7b26eac8797ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.813ex; height:2.176ex;" alt="{\displaystyle 15\cdot 3.}"> Розкласти на множники можна й поліном. Розкласти поліном на множники - означає представити його у вигляді добутку двох або декількох поліномів. Наприклад, у поліномі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy+xz}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy+xz}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ace25b83c281ad54483fe859b73cf3a7ffd874e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.743ex; height:2.343ex;" alt="{\displaystyle xy+xz}"> можна виділити спільний множник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> суми <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y+z,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y+z,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8619bcc2735130f2cc30de94e188910c6dea8530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.731ex; height:2.343ex;" alt="{\displaystyle y+z,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(y+z)=xy+xz.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(y+z)=xy+xz.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/505b66d68d968274eb70240fa598f138788b8508" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.712ex; height:2.843ex;" alt="{\displaystyle x(y+z)=xy+xz.}"> Тут ми скористалися розподільною властивістю множення і винесли спільний множник членів полінома за дужки. Розгляньмо декілька прикладів: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y-30x^{2}y^{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y-30x^{2}y^{2};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0704659c1cf8f7cbf589277eb3a0694b1751f0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.113ex; height:3.009ex;" alt="{\displaystyle 5x^{4}y-30x^{2}y^{2};}"> спочатку віднайдемо найбільший спільний дільник абсолютних величин коефіцієнтів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29483407999b8763f0ea335cf715a6a5e809f44b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 5}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-30),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>30</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-30),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e29ae143254759dd78827a2cc649578ab70448a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.589ex; height:2.843ex;" alt="{\displaystyle (-30),}"> зокрема <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|5|;\,|-30|)=5;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>5</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>30</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|5|;\,|-30|)=5;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28e362ab3178dd34d17991d389a088bb1a512035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.303ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|5|;\,|-30|)=5;}"> степінь <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/901fc910c19990d0dbaaefe4726ceb1a4e217a0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 2}"> із основою <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> входить в якості множника до обох членів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y-30x^{2}y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y-30x^{2}y^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6833fee77914f91e5a6e75ebcdec7f826a4da36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.466ex; height:3.009ex;" alt="{\displaystyle 5x^{4}y-30x^{2}y^{2}}"> даного поліному, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y-30x^{2}y^{2}=x^{2}(5x^{2}y-30y^{2});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y-30x^{2}y^{2}=x^{2}(5x^{2}y-30y^{2});}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd2010f530f68e0b3b578e07e30dc34108f79fb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.486ex; height:3.176ex;" alt="{\displaystyle 5x^{4}y-30x^{2}y^{2}=x^{2}(5x^{2}y-30y^{2});}"> степінь <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> із основою <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> також входить в якості множника до членів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9deecf76c3d90ccd3b3901e3f4d1f9363ea53649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.702ex; height:3.009ex;" alt="{\displaystyle 5x^{4}y}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -30x^{2}y^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -30x^{2}y^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0662be9fcbc86ab78aba076f2718a7a39439db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.379ex; height:3.009ex;" alt="{\displaystyle -30x^{2}y^{2},}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y-30x^{2}y^{2}=y(5x^{4}-30x^{2}y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y-30x^{2}y^{2}=y(5x^{4}-30x^{2}y);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b90c811f1860a12f520dd08f6b4d7d479e090c77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.427ex; height:3.176ex;" alt="{\displaystyle 5x^{4}y-30x^{2}y^{2}=y(5x^{4}-30x^{2}y);}"> отже, за дужки можна винести моном <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{2}y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{2}y,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/624dab7c4fafe6bf21de34dc420715dd73c5b7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.349ex; height:3.009ex;" alt="{\displaystyle 5x^{2}y,}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5x^{4}y-30x^{2}y^{2}=5x^{2}\cdot x^{2}\cdot y-6x^{2}y\cdot 5y=5x^{2}y(5x^{2}-6y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mi>y</mi> <mo>−</mo> <mn>30</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⋅</mo> <mi>y</mi> <mo>−</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>⋅</mo> <mn>5</mn> <mi>y</mi> <mo>=</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5x^{4}y-30x^{2}y^{2}=5x^{2}\cdot x^{2}\cdot y-6x^{2}y\cdot 5y=5x^{2}y(5x^{2}-6y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe4df604bd10eb83704222900fda61ffb3199f36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.509ex; height:3.176ex;" alt="{\displaystyle 5x^{4}y-30x^{2}y^{2}=5x^{2}\cdot x^{2}\cdot y-6x^{2}y\cdot 5y=5x^{2}y(5x^{2}-6y).}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8+4x;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8+4x;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5be9cc24bfdb4732eb3c70d88273bbbda4e1c56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.464ex; height:2.509ex;" alt="{\displaystyle 8+4x;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|8|;|4|)=4;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|8|;|4|)=4;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23844ea67a789049010a8620ec5f7e731d108086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.236ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|8|;|4|)=4;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8+4x=4(2+x);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8+4x=4(2+x);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a734c43521f4bf58e2a7e629178357b83f36a11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.545ex; height:2.843ex;" alt="{\displaystyle 8+4x=4(2+x);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15x-10;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>10</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15x-10;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1ddff8f7f4300bc558873099b011dc2364735a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.789ex; height:2.509ex;" alt="{\displaystyle 15x-10;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|15|;|-10|)=5;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|15|;|-10|)=5;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa30ebda068e19c3ff48d14ec313623c9d206161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.401ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|15|;|-10|)=5;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15x-10=5(3x-2);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>10</mn> <mo>=</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15x-10=5(3x-2);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7be1d4d17a595fa181ec27e424269ae2c27484e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.032ex; height:2.843ex;" alt="{\displaystyle 15x-10=5(3x-2);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+3xy;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+3xy;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17265e95906bcc2972be6820599469bf881bab2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.95ex; height:2.509ex;" alt="{\displaystyle 3x+3xy;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|3|;|3|)=3;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|3|;|3|)=3;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4423b359143d05509ba4464acf39bff83391f128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.236ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|3|;|3|)=3;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+3xy=3x(1+y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+3xy=3x(1+y);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b6aa62f09bd9086cb4d5ba9f64bc486d0bd7c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.185ex; height:2.843ex;" alt="{\displaystyle 3x+3xy=3x(1+y);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15x-25y;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>25</mn> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15x-25y;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a88bacb6c50f23f95ecc29f9c40122083511dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.945ex; height:2.509ex;" alt="{\displaystyle 15x-25y;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|15|;|-25|)=5;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>15</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>25</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|15|;|-25|)=5;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73ef404d30cf74578532770a78de75c59e9d92db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.401ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|15|;|-25|)=5;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15x-25y=5(3x-5y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mi>x</mi> <mo>−</mo> <mn>25</mn> <mi>y</mi> <mo>=</mo> <mn>5</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15x-25y=5(3x-5y);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6541aed0fecc60a5cd446fd528cf7602890ad27f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.343ex; height:2.843ex;" alt="{\displaystyle 15x-25y=5(3x-5y);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle am+km=m(a+k);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>m</mi> <mo>+</mo> <mi>k</mi> <mi>m</mi> <mo>=</mo> <mi>m</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle am+km=m(a+k);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298e1b39cf7dcc563637808c28a87e949b25f89a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.239ex; height:2.843ex;" alt="{\displaystyle am+km=m(a+k);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8xz-6xy;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mi>z</mi> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>y</mi> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8xz-6xy;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c86742ed8fb380f71b638c17b9e8f83dfb8808a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.038ex; height:2.509ex;" alt="{\displaystyle 8xz-6xy;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|8|;|-6|)=2;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>8</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|8|;|-6|)=2;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3105a478ec192eabdb0bc1ee45a3fb0b436ff0cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:22.076ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|8|;|-6|)=2;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8xz-6xy=2x(4z-3y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> <mi>x</mi> <mi>z</mi> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mi>z</mi> <mo>−</mo> <mn>3</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8xz-6xy=2x(4z-3y);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/035ebb8e0acf9d961a2af4b9c1936b8e60aa7301" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.524ex; height:2.843ex;" alt="{\displaystyle 8xz-6xy=2x(4z-3y);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -24x-18x^{3};\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>24</mn> <mi>x</mi> <mo>−</mo> <mn>18</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -24x-18x^{3};\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b8ccacd24815aea2efe76d1fee69d75dc315d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.981ex; height:3.009ex;" alt="{\displaystyle -24x-18x^{3};\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|-24|;|-18|)=6;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>24</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|-24|;|-18|)=6;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dea1aa8fbbfb14a376d1cbd0cc68861f7343d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.242ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|-24|;|-18|)=6;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -24-18x^{3}=6x(-4-3x^{2});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>24</mn> <mo>−</mo> <mn>18</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>6</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>4</mn> <mo>−</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -24-18x^{3}=6x(-4-3x^{2});}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f04bf0cff56ff88c52e252bc60e357a25f78ed95" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.086ex; height:3.176ex;" alt="{\displaystyle -24-18x^{3}=6x(-4-3x^{2});}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -9x^{5}-27x^{3}+18x^{4};\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -9x^{5}-27x^{3}+18x^{4};\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bbc386c2872e85c4330e0d32300af466723acc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.422ex; height:3.009ex;" alt="{\displaystyle -9x^{5}-27x^{3}+18x^{4};\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{НСД}}(|-9|;|-27|;|18|)=9;\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>НСД</mtext> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>9</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <mn>27</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>18</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{НСД}}(|-9|;|-27|;|18|)=9;\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daa8c066a225696ab103c45ba0f9e08ce5eea11f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:30.732ex; height:3.343ex;" alt="{\displaystyle {\text{НСД}}(|-9|;|-27|;|18|)=9;\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -9x^{5}-27x^{3}+18x^{4}=9x^{3}(-x^{2}+2x-3);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <mn>27</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -9x^{5}-27x^{3}+18x^{4}=9x^{3}(-x^{2}+2x-3);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a4bdd77938d2f8b59f2e901fe3d9d18f03ab49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.081ex; height:3.176ex;" alt="{\displaystyle -9x^{5}-27x^{3}+18x^{4}=9x^{3}(-x^{2}+2x-3);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{4}-a^{3}-a^{2};\quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{4}-a^{3}-a^{2};\quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4330a617246b433f3d91fddf842807f90403070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.665ex; height:3.009ex;" alt="{\displaystyle 2a^{4}-a^{3}-a^{2};\quad }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2a^{4}-a^{3}-a^{2}=a^{2}(2a^{2}-a-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2a^{4}-a^{3}-a^{2}=a^{2}(2a^{2}-a-1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af3be8bc14882bb4c529895569e2ee061462e440" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.054ex; height:3.176ex;" alt="{\displaystyle 2a^{4}-a^{3}-a^{2}=a^{2}(2a^{2}-a-1).}"></li></ul> Розкладати поліном на множники можна й способом групування, який полягає у перегрупуванні доданків полінома й подальшому об'єднанні у групи таким чином, щоб після винесення (якщо це можливо) спільного множника з кожного доданка у даній групі у дужках отримувався вираз, який, у свою чергу, є спільним множником вже для кожної групи. Розгляньмо добуток поліномів <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-b)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de142beb9731d08ae0cf1d37fbb1e7bc6ccd4dc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.877ex; height:2.843ex;" alt="{\displaystyle (a-b)}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a12281576b60165fa760776dd09f11bb675c6e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.782ex; height:2.843ex;" alt="{\displaystyle (x+y),}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a-b)(x+y)=ax+ay-bx-by.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-b)(x+y)=ax+ay-bx-by.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340c24b78c5d03c662b712f2f4fd216f33abe5f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.703ex; height:2.843ex;" alt="{\displaystyle (a-b)(x+y)=ax+ay-bx-by.}"> У отриманому поліномі члени <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d44662eeb8cbba7277da838b75c77d8cd3a4547" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.559ex; height:1.676ex;" alt="{\displaystyle ax}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ay}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ay}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e02a91639ea7af4384c0a96daad308ac12941770" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.009ex;" alt="{\displaystyle ay}"> мають спільний множник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10b8c6805aa963795f06f1fcb259463f63d56412" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.877ex; height:2.009ex;" alt="{\displaystyle a;}"> члени <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -bx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>b</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -bx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128ee5f5686c97c419cbc49a0447021e538aaf77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.135ex; height:2.343ex;" alt="{\displaystyle -bx}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -by}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>b</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -by}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17add19ee0ae0e91212ed8e89ee837bc565a5899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.961ex; height:2.509ex;" alt="{\displaystyle -by}"> мають спільний множник <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-b).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d0ddf267e7dbae3ae3de84aeeb95de2b5d78f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.262ex; height:2.843ex;" alt="{\displaystyle (-b).}"> Таким чином, маємо дві групи членів полінома: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+ay}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+ay}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a94bab3c21025dd7440f668ad4452cb5e73bac90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.785ex; height:2.343ex;" alt="{\displaystyle ax+ay}"> та <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -bx-by.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -bx-by.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00e3c9fc10da4b98f2d3fcd455ac3496dd47a5ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.776ex; height:2.509ex;" alt="{\displaystyle -bx-by.}"> У кожній групі можна винести спільний множник за дужки, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(x+y)-b(x+y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(x+y)-b(x+y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d47181bd5087a79446f7fb51a726b508cce654c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.984ex; height:2.843ex;" alt="{\displaystyle a(x+y)-b(x+y).}"> Розгляньмо перетворення у зворотному порядку: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax+ay-bx-by=(ax+ay)-(bx+by)=a(x+y)-b(x+y)=(x+y)(a-b).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> <mo>−</mo> <mi>b</mi> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax+ay-bx-by=(ax+ay)-(bx+by)=a(x+y)-b(x+y)=(x+y)(a-b).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6749f09f0d342142020b0a2e5d58c61b48a5bf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:82.803ex; height:2.843ex;" alt="{\displaystyle ax+ay-bx-by=(ax+ay)-(bx+by)=a(x+y)-b(x+y)=(x+y)(a-b).}"> Описаний спосіб розкладу полінома на множники називається способом групування. Застосовуючи цей спосіб, необхідно групувати члени таким чином, щоб з отриманих груп можна було виділити спільний множник. Розгляньмо декілька прикладів: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+b)(x-y)=ax-ay+bx-by=a(x-y)+b(x-y);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mi>y</mi> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mi>y</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)(x-y)=ax-ay+bx-by=a(x-y)+b(x-y);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7b9c437e16f533924dfe5350727102eaee165be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.139ex; height:2.843ex;" alt="{\displaystyle (a+b)(x-y)=ax-ay+bx-by=a(x-y)+b(x-y);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3xy+12by+9x-4by^{2}=(3xy+9x)-(4by^{2}+12by)=3x(y+3)-4by(y+3)=(y+3)(3x-4by);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>12</mn> <mi>b</mi> <mi>y</mi> <mo>+</mo> <mn>9</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mi>b</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>9</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>b</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>b</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>4</mn> <mi>b</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mi>b</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3xy+12by+9x-4by^{2}=(3xy+9x)-(4by^{2}+12by)=3x(y+3)-4by(y+3)=(y+3)(3x-4by);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e730c887208fb416e8e87f849cd0cc6d819a64c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:100.596ex; height:3.176ex;" alt="{\displaystyle 3xy+12by+9x-4by^{2}=(3xy+9x)-(4by^{2}+12by)=3x(y+3)-4by(y+3)=(y+3)(3x-4by);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}-5a+6;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <mi>a</mi> <mo>+</mo> <mn>6</mn> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-5a+6;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10d7abde8302291cbf0ff6666644e3ca29cfccbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.166ex; height:3.009ex;" alt="{\displaystyle a^{2}-5a+6;}"> подамо член <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -5a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>5</mn> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -5a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40bc84157677b89c4eed96fcf7820bde2ef7be04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.2ex; height:2.343ex;" alt="{\displaystyle -5a}"> у вигляді різниці <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -3a-2a;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -3a-2a;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd8adc89ca82f982ded58110841d1de12ccada68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.08ex; height:2.509ex;" alt="{\displaystyle -3a-2a;}"> тоді <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}-3a-2a+6=a(a-3)-2(a-3)=(a-3)(a-2);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-3a-2a+6=a(a-3)-2(a-3)=(a-3)(a-2);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58eded7164fa12ec13f540b5216921be87b18f53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.997ex; height:3.176ex;" alt="{\displaystyle a^{2}-3a-2a+6=a(a-3)-2(a-3)=(a-3)(a-2);}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{3}y(-5y^{4}+2xy)=-10x^{3}y^{5}+4x^{4}y^{2};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo>−</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{3}y(-5y^{4}+2xy)=-10x^{3}y^{5}+4x^{4}y^{2};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e2665ace09ee9c3c906943434a227a61e6a3caf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.263ex; height:3.176ex;" alt="{\displaystyle 2x^{3}y(-5y^{4}+2xy)=-10x^{3}y^{5}+4x^{4}y^{2};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2x+y-3z)\cdot (-5yz^{2})=10xyz^{2}-5y^{2}z^{2}-15yz^{3};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>3</mn> <mi>z</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> <mi>x</mi> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>15</mn> <mi>y</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2x+y-3z)\cdot (-5yz^{2})=10xyz^{2}-5y^{2}z^{2}-15yz^{3};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16fe484b2646a71022bfba6c13a26f23445474e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.675ex; height:3.176ex;" alt="{\displaystyle (2x+y-3z)\cdot (-5yz^{2})=10xyz^{2}-5y^{2}z^{2}-15yz^{3};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -4k^{3}n(kn^{2}-kn-n^{2})-(2mn)^{3}\cdot ((m^{5n+1}n^{2})^{8})^{0}=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3}\cdot 1=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>n</mi> <mo stretchy="false">(</mo> <mi>k</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>k</mi> <mi>n</mi> <mo>−</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>m</mi> <mi>n</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mo>−</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <msup> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -4k^{3}n(kn^{2}-kn-n^{2})-(2mn)^{3}\cdot ((m^{5n+1}n^{2})^{8})^{0}=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3}\cdot 1=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13712bb8182d20252da6a9b22ce1c337186df14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:128.191ex; height:3.176ex;" alt="{\displaystyle -4k^{3}n(kn^{2}-kn-n^{2})-(2mn)^{3}\cdot ((m^{5n+1}n^{2})^{8})^{0}=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3}\cdot 1=-4k^{4}n^{3}+4k^{4}n^{2}+4k^{3}n^{3}-8m^{3}n^{3};}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}b\cdot (1+2a+b^{3})-(a^{3}b^{3}+b^{4})=a^{2}b+2a^{3}b+a^{2}b^{4}-a^{3}b^{3}-b^{4}=a^{2}b^{4}-a^{3}b^{3}-b^{4}+2a^{3}b+a^{2}b;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}b\cdot (1+2a+b^{3})-(a^{3}b^{3}+b^{4})=a^{2}b+2a^{3}b+a^{2}b^{4}-a^{3}b^{3}-b^{4}=a^{2}b^{4}-a^{3}b^{3}-b^{4}+2a^{3}b+a^{2}b;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe79c18ba0df451e4cd47773d3feb455588cacb4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:98.4ex; height:3.176ex;" alt="{\displaystyle a^{2}b\cdot (1+2a+b^{3})-(a^{3}b^{3}+b^{4})=a^{2}b+2a^{3}b+a^{2}b^{4}-a^{3}b^{3}-b^{4}=a^{2}b^{4}-a^{3}b^{3}-b^{4}+2a^{3}b+a^{2}b;}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)...(t-x)(t-y)(t-z);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)...(t-x)(t-y)(t-z);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0867787d5e921439ea8329b4c930ea79a269468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.538ex; height:2.843ex;" alt="{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)...(t-x)(t-y)(t-z);}"> тут перемножуються послідовно різниці <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"> та змінних, позначених символами латинської абетки. Варто звернути увагу, що серед множників є також й <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t-t)=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t-t)=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5d3b867c9d2e1d2a8c722400515e67fa822690d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.237ex; height:2.843ex;" alt="{\displaystyle (t-t)=0,}"> внаслідок чого весь добуток перетворюється на нуль. Таким чином, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)\cdot \cdot \cdot (t-x)(t-y)(t-z)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>e</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo>⋅</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)\cdot \cdot \cdot (t-x)(t-y)(t-z)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc02c8d5827ae8242cf99f6d4359a51afc50ec5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:68.702ex; height:2.843ex;" alt="{\displaystyle (t-a)(t-b)(t-c)(t-d)(t-e)(t-f)\cdot \cdot \cdot (t-x)(t-y)(t-z)=0.}"></li> <li>Нехай деяке число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> при діленні на 6 діє в остачі 4, а друге число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> при діленні на 6 в остачі дає 3. Необхідно довести, що їх добуток, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7a2b9c2d667b8ef76a572e3068f9740493152de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.553ex; height:2.509ex;" alt="{\displaystyle a\cdot b,}"> ділиться на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"> без остачі. Зрозуміло, що <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=6k+4,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>6</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=6k+4,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a87c0ac3054cd526d244efe36bc22dc31faf93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.352ex; height:2.509ex;" alt="{\displaystyle a=6k+4,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> - неповна частка від ділення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:6,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>:</mo> <mn>6</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a:6,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc3cfdbd83a1b81cfd29da104035f8efd7de06f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.976ex; height:2.509ex;" alt="{\displaystyle a:6,}"> а <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b=6l+3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>=</mo> <mn>6</mn> <mi>l</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b=6l+3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56db706a5796ca085c77cf1da1873c2df71241c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.601ex; height:2.509ex;" alt="{\displaystyle b=6l+3,}"> де <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle l}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829091f745070b9eb97a80244129025440a1cfac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}"> - неповна частка від ділення <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b:6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>:</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b:6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9ba7444b92928f10b02f3492aa74ef089b3097e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.744ex; height:2.176ex;" alt="{\displaystyle b:6.}"> Розгляньмо добуток <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (6k+4)(6l+3)=36kl+18k+24l+12.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>6</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>l</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>36</mn> <mi>k</mi> <mi>l</mi> <mo>+</mo> <mn>18</mn> <mi>k</mi> <mo>+</mo> <mn>24</mn> <mi>l</mi> <mo>+</mo> <mn>12.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (6k+4)(6l+3)=36kl+18k+24l+12.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c3a0532fb5e131c215e7afef6daa99946932337" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.229ex; height:2.843ex;" alt="{\displaystyle (6k+4)(6l+3)=36kl+18k+24l+12.}"> Члени полінома <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36kl+18k+24l+12}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mi>k</mi> <mi>l</mi> <mo>+</mo> <mn>18</mn> <mi>k</mi> <mo>+</mo> <mn>24</mn> <mi>l</mi> <mo>+</mo> <mn>12</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36kl+18k+24l+12}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd330aefb3d5c455eff82304f7dae23eb501ece8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:21.63ex; height:2.343ex;" alt="{\displaystyle 36kl+18k+24l+12}"> містять коефіцієнти <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 36,\,18,\,24,\,12,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>18</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>24</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>12</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36,\,18,\,24,\,12,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2ca7b7a8c80e26cf0d7388d7332b9fcaa39e85c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.21ex; height:2.509ex;" alt="{\displaystyle 36,\,18,\,24,\,12,}"> які діляться націло на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641abae36403783c71568aae30295e0e32b72daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 6.}"> Ми можемо винести число <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d81124420a058a7474dfeda48228fb6ee1e253" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 6}"> за дужки: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6(6kl+3k+4l+2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mo stretchy="false">(</mo> <mn>6</mn> <mi>k</mi> <mi>l</mi> <mo>+</mo> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mn>4</mn> <mi>l</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6(6kl+3k+4l+2).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d1fdf9eb4b4364202b8710223a403aac19ac2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.599ex; height:2.843ex;" alt="{\displaystyle 6(6kl+3k+4l+2).}"> Це означає, що добуток <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"> ділиться на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/641abae36403783c71568aae30295e0e32b72daa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.809ex; height:2.176ex;" alt="{\displaystyle 6.}"></li> <li>Нехай довжина прямокутника на 4 см більша за ширину. Якщо його довжину зменшити на 1 см, а ширину збільшити на 2 см, то площа стане на 10 см<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d19a0bee8c4bfc83c25449369b10341886e6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.054ex; height:2.509ex;" alt="{\displaystyle {}^{2}}"> більшою. Необхідно віднайти розміри зміненого прямокутника. За умовою задачі, його ширина складає <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb96677ba71b937617ca8751955f884f6306b64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.644ex; height:2.509ex;" alt="{\displaystyle b,}"> а довжина на 4 більша від ширини: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b+4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=b+4.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/574db0e2472b8f787487baae77eeeff525127c43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.976ex; height:2.343ex;" alt="{\displaystyle a=b+4.}"> Площа (в загальному, площа прямокутника обчислюється як добуток довжини і ширини <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=a\cdot b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9c1ae41a3a97b5444e169bb6fcc24aa73c4a12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.504ex; height:2.176ex;" alt="{\displaystyle S=a\cdot b}">) у такому випадку буде становити <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}=b(b+4).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}=b(b+4).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b73cd4b02a8ae7b4ad99f53b773a5a603efc5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.032ex; height:2.843ex;" alt="{\displaystyle S_{1}=b(b+4).}"> Зменшимо його довжину на <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"> см, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+4-1=b+3,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mn>4</mn> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+4-1=b+3,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f327a42e5f6601aa20a3e252815f1dd1c9c6e88b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.749ex; height:2.509ex;" alt="{\displaystyle b+4-1=b+3,}"> а ширину збільшимо на 2 см, тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b+2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fb3991d1d8ebe64846729dfe5d104e2862b8382" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5ex; height:2.343ex;" alt="{\displaystyle b+2}"> см. Нова площа становитиме <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}=(b+3)(b+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}=(b+3)(b+2)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43767d525ecbef0a037c159b7b98bd9c80bd78bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.197ex; height:2.843ex;" alt="{\displaystyle S_{2}=(b+3)(b+2)}"> см<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466beb0513b5df0f97c84c23343c93b95baa41bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.701ex; height:3.009ex;" alt="{\displaystyle {}^{2},}"> що на 10 см<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d19a0bee8c4bfc83c25449369b10341886e6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.054ex; height:2.509ex;" alt="{\displaystyle {}^{2}}"> більше від площі <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b(b+4),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b(b+4),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b568225aaca05193e2c2ca2ddab6a3289cdcc8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.454ex; height:2.843ex;" alt="{\displaystyle b(b+4),}"> тобто <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}-S_{1}=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}-S_{1}=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33648aa7f726bbfa4ff45079cabf9eb3fa72fa5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.222ex; height:2.509ex;" alt="{\displaystyle S_{2}-S_{1}=10}"> см<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {}^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {}^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbaac89c4d1af1773776bcafe0fd474d5e38e5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.701ex; height:2.676ex;" alt="{\displaystyle {}^{2}.}"> Таким чином, можна записати <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (b+3)(b+2)-b(b+4)=10\Rightarrow b^{2}+2b+3b+6-b^{2}-b4=10\Rightarrow b+6=10\Rightarrow b=10-6\Rightarrow b=4.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>b</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> <mo stretchy="false">⇒</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mi>b</mi> <mo>+</mo> <mn>6</mn> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <mn>4</mn> <mo>=</mo> <mn>10</mn> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mn>10</mn> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>=</mo> <mn>10</mn> <mo>−</mo> <mn>6</mn> <mo stretchy="false">⇒</mo> <mi>b</mi> <mo>=</mo> <mn>4.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b+3)(b+2)-b(b+4)=10\Rightarrow b^{2}+2b+3b+6-b^{2}-b4=10\Rightarrow b+6=10\Rightarrow b=10-6\Rightarrow b=4.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b42cc9b1bd4baecac55e77ab19e4ae5801d83d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:102.271ex; height:3.176ex;" alt="{\displaystyle (b+3)(b+2)-b(b+4)=10\Rightarrow b^{2}+2b+3b+6-b^{2}-b4=10\Rightarrow b+6=10\Rightarrow b=10-6\Rightarrow b=4.}"> Таким чином, довжина прямокутника складає <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aaa997e6ad67716cfaa9a02c4df860bf60a95b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 8}"> см, ширина <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/295b4bf1de7cd3500e740e0f4f0635db22d87b42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 4}"> см.</li></ul> ← <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Натуральні числа">Натуральні числа</a> · <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Раціональні числа">Раціональні числа</a> →<div id="bottom-navigation" style="float: none; text-align: center; border-style:outset; border-width:3px; border-color:green;" class="noprint">← <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%9D%D0%B0%D1%82%D1%83%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Натуральні числа">Натуральні числа</a> · <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8" title="Основні числові системи">Зміст</a> · <a href="/wiki/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A0%D0%B0%D1%86%D1%96%D0%BE%D0%BD%D0%B0%D0%BB%D1%8C%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0" title="Основні числові системи/Раціональні числа">Раціональні числа</a> →</div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Отримано з <a dir="ltr" href="https://uk.wikibooks.org/w/index.php?title=Основні_числові_системи/Цілі_числа&oldid=36331">https://uk.wikibooks.org/w/index.php?title=Основні_числові_системи/Цілі_числа&oldid=36331</a></div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/%D0%A1%D0%BF%D0%B5%D1%86%D1%96%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0:%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%97" title="Спеціальна:Категорії">Категорія</a>: <ul><li><a href="/wiki/%D0%9A%D0%B0%D1%82%D0%B5%D0%B3%D0%BE%D1%80%D1%96%D1%8F:%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8" title="Категорія:Основні числові системи">Основні числові системи</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> Цю сторінку востаннє відредаговано о 22:03, 28 січня 2022.</li> <li id="footer-info-copyright">Текст доступний на умовах ліцензії <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.uk">Creative Commons Attribution-ShareAlike</a>; також можуть діяти додаткові умови. Детальніше див. <a class="external text" href="https://foundation.wikimedia.org/wiki/Policy:Terms_of_Use/uk">Умови використання</a>.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Політика конфіденційності</a></li> <li id="footer-places-about"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D1%96%D0%B4%D1%80%D1%83%D1%87%D0%BD%D0%B8%D0%BA:%D0%9F%D1%80%D0%BE">Про Вікіпідручник</a></li> <li id="footer-places-disclaimers"><a href="/wiki/%D0%92%D1%96%D0%BA%D1%96%D0%BF%D1%96%D0%B4%D1%80%D1%83%D1%87%D0%BD%D0%B8%D0%BA:%D0%92%D1%96%D0%B4%D0%BC%D0%BE%D0%B2%D0%B0_%D0%B2%D1%96%D0%B4_%D0%B2%D1%96%D0%B4%D0%BF%D0%BE%D0%B2%D1%96%D0%B4%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D1%81%D1%82%D1%96">Відмова від відповідальності</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Кодекс поведінки</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Розробники</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/uk.wikibooks.org">Статистика</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Куки</a></li> <li id="footer-places-mobileview"><a href="//uk.m.wikibooks.org/w/index.php?title=%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Мобільний вигляд</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-7dfb9d98f5-6hzmm","wgBackendResponseTime":221,"wgPageParseReport":{"limitreport":{"cputime":"0.416","walltime":"0.668","ppvisitednodes":{"value":2274,"limit":1000000},"postexpandincludesize":{"value":2138,"limit":2097152},"templateargumentsize":{"value":486,"limit":2097152},"expansiondepth":{"value":5,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":18216,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 22.173 1 -total"," 17.76% 3.938 1 Шаблон:Гортання_сторінок"," 17.75% 3.936 1 Шаблон:Вікіпедія"]},"cachereport":{"origin":"mw-web.codfw.main-7c54859c9b-jvzxp","timestamp":"20241111145945","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u041e\u0441\u043d\u043e\u0432\u043d\u0456 \u0447\u0438\u0441\u043b\u043e\u0432\u0456 \u0441\u0438\u0441\u0442\u0435\u043c\u0438\/\u0426\u0456\u043b\u0456 \u0447\u0438\u0441\u043b\u0430","url":"https:\/\/uk.wikibooks.org\/wiki\/%D0%9E%D1%81%D0%BD%D0%BE%D0%B2%D0%BD%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%BE%D0%B2%D1%96_%D1%81%D0%B8%D1%81%D1%82%D0%B5%D0%BC%D0%B8\/%D0%A6%D1%96%D0%BB%D1%96_%D1%87%D0%B8%D1%81%D0%BB%D0%B0","sameAs":"http:\/\/www.wikidata.org\/entity\/Q12503","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q12503","author":{"@type":"Organization","name":"\u0423\u0447\u0430\u0441\u043d\u0438\u043a\u0438 \u043f\u0440\u043e\u0435\u043a\u0442\u0456\u0432 \u0412\u0456\u043a\u0456\u043c\u0435\u0434\u0456\u0430"},"publisher":{"@type":"Organization","name":"\u0424\u043e\u043d\u0434 \u0412\u0456\u043a\u0456\u043c\u0435\u0434\u0456\u0430","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2022-01-28T21:16:24Z","dateModified":"2022-01-28T22:03:56Z","headline":"\u043c\u043d\u043e\u0436\u0438\u043d\u0430 \u0447\u0438\u0441\u0435\u043b"}</script> </body> </html>

CINXE.COM

Основні числові системи/Цілі числа — Вікіпідручник