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data-file-height=\"156\" /\u003E\u003C/a\u003E\u003C/span\u003E\u003C/td\u003E\u003Ctd class=\"mbox-text\"\u003EPišite i uređujte članke u \u003Cb\u003E\u003Ca href=\"/wiki/Wikipedija:Wikiprojekt_Hrvatska/1100._obljetnica_Hrvatskog_Kraljevstva\" title=\"Wikipedija:Wikiprojekt Hrvatska/1100. obljetnica Hrvatskog Kraljevstva\"\u003Ewikimaratonu povodom 1100. obljetnice Hrvatskoga Kraljevstva\u003C/a\u003E\u003C/b\u003E.\u003C/td\u003E\u003Ctd class=\"mbox-imageright\"\u003E\u003Cspan typeof=\"mw:File\"\u003E\u003Ca href=\"/wiki/Datoteka:Pleter12.svg\" class=\"mw-file-description\"\u003E\u003Cimg src=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Pleter12.svg/60px-Pleter12.svg.png\" decoding=\"async\" width=\"60\" height=\"18\" class=\"mw-file-element\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Pleter12.svg/90px-Pleter12.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/Pleter12.svg/120px-Pleter12.svg.png 2x\" data-file-width=\"156\" 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komentare/Nazivi članaka\"\u003E\u003Cb\u003EZahtjev za komentare o nazivima članaka\u003C/b\u003E\u003C/a\u003E\u003C/td\u003E\u003C/tr\u003E\u003C/tbody\u003E\u003C/table\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E\u003C/div\u003E";}}());</script></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="Mrežno mjesto"> <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="Sadržaj" 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">Sadržaj</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">sakrij</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">Početak</div> </a> </li> <li id="toc-Monomi,_binomi,_trinomi,_itd." class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Monomi,_binomi,_trinomi,_itd."> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Monomi, binomi, trinomi, itd.</span> </div> </a> <ul id="toc-Monomi,_binomi,_trinomi,_itd.-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nul-polinom" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nul-polinom"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Nul-polinom</span> </div> </a> <ul id="toc-Nul-polinom-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Računske_operacije_s_polinomima" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Računske_operacije_s_polinomima"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Računske operacije s polinomima</span> </div> </a> <button aria-controls="toc-Računske_operacije_s_polinomima-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Sadržaj cjeline Računske operacije s polinomima</span> </button> <ul id="toc-Računske_operacije_s_polinomima-sublist" class="vector-toc-list"> <li id="toc-Primjeri" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primjeri"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Primjeri</span> </div> </a> <ul id="toc-Primjeri-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Uporaba_polinoma" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Uporaba_polinoma"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Uporaba polinoma</span> </div> </a> <ul id="toc-Uporaba_polinoma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nultočke_polinoma" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Nultočke_polinoma"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Nultočke polinoma</span> </div> </a> <ul id="toc-Nultočke_polinoma-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Izvori" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Izvori"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Izvori</span> </div> </a> <ul id="toc-Izvori-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="Sadržaj" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Sadržaj" > <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="Prikaz sadržaja stranice" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Prikaz sadržaja stranice</span> </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"><span class="mw-page-title-main">Polinom</span></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="Idi na druge jezične varijante članka. Dostupan je na 82 jezika" > <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-82" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">82 jezika</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Polinoom" title="Polinoom – afrikaans" lang="af" hreflang="af" data-title="Polinoom" data-language-autonym="Afrikaans" data-language-local-name="afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AA%D8%B9%D8%AF%D8%AF%D8%A9_%D8%A7%D9%84%D8%AD%D8%AF%D9%88%D8%AF" title="متعددة الحدود – arapski" lang="ar" hreflang="ar" data-title="متعددة الحدود" data-language-autonym="العربية" data-language-local-name="arapski" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Polinomiu" title="Polinomiu – asturijski" lang="ast" hreflang="ast" data-title="Polinomiu" data-language-autonym="Asturianu" data-language-local-name="asturijski" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C3%87oxh%C9%99dli" title="Çoxhədli – azerbajdžanski" lang="az" hreflang="az" data-title="Çoxhədli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbajdžanski" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%9A%D2%AF%D0%BF%D0%B1%D1%8B%D1%83%D1%8B%D0%BD" title="Күпбыуын – baškirski" lang="ba" hreflang="ba" data-title="Күпбыуын" data-language-autonym="Башҡортса" data-language-local-name="baškirski" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B0%D0%B3%D0%B0%D1%87%D0%BB%D0%B5%D0%BD" title="Мнагачлен – bjeloruski" lang="be" hreflang="be" data-title="Мнагачлен" data-language-autonym="Беларуская" data-language-local-name="bjeloruski" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%BD%D0%B0%D0%B3%D0%B0%D1%81%D0%BA%D0%BB%D0%B0%D0%B4" title="Мнагасклад – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Мнагасклад" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – bugarski" lang="bg" hreflang="bg" data-title="Многочлен" data-language-autonym="Български" data-language-local-name="bugarski" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A6%B9%E0%A7%81%E0%A6%AA%E0%A6%A6%E0%A7%80" title="বহুপদী – bangla" lang="bn" hreflang="bn" data-title="বহুপদী" data-language-autonym="বাংলা" data-language-local-name="bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Polinom" title="Polinom – bosanski" lang="bs" hreflang="bs" data-title="Polinom" data-language-autonym="Bosanski" data-language-local-name="bosanski" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Polinomi" title="Polinomi – katalonski" lang="ca" hreflang="ca" data-title="Polinomi" data-language-autonym="Català" data-language-local-name="katalonski" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%DA%95%D8%A7%D8%AF%DB%95%D8%AF%D8%A7%D8%B1" title="ڕادەدار – soranski kurdski" lang="ckb" hreflang="ckb" data-title="ڕادەدار" data-language-autonym="کوردی" data-language-local-name="soranski kurdski" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Polynom" title="Polynom – češki" lang="cs" hreflang="cs" data-title="Polynom" data-language-autonym="Čeština" data-language-local-name="češki" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – čuvaški" lang="cv" hreflang="cv" data-title="Полином" data-language-autonym="Чӑвашла" data-language-local-name="čuvaški" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Polynomial" title="Polynomial – velški" lang="cy" hreflang="cy" data-title="Polynomial" data-language-autonym="Cymraeg" data-language-local-name="velški" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Polynomium" title="Polynomium – danski" lang="da" hreflang="da" data-title="Polynomium" data-language-autonym="Dansk" data-language-local-name="danski" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Polynom" title="Polynom – njemački" lang="de" hreflang="de" data-title="Polynom" data-language-autonym="Deutsch" data-language-local-name="njemački" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-diq mw-list-item"><a href="https://diq.wikipedia.org/wiki/Polinom" title="Polinom – Dimli" lang="diq" hreflang="diq" data-title="Polinom" data-language-autonym="Zazaki" data-language-local-name="Dimli" class="interlanguage-link-target"><span>Zazaki</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%BF%CE%BB%CF%85%CF%8E%CE%BD%CF%85%CE%BC%CE%BF" title="Πολυώνυμο – grčki" lang="el" hreflang="el" data-title="Πολυώνυμο" data-language-autonym="Ελληνικά" data-language-local-name="grčki" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Polynomial" title="Polynomial – engleski" lang="en" hreflang="en" data-title="Polynomial" data-language-autonym="English" data-language-local-name="engleski" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Polinomo" title="Polinomo – esperanto" lang="eo" hreflang="eo" data-title="Polinomo" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Polinomio" title="Polinomio – španjolski" lang="es" hreflang="es" data-title="Polinomio" data-language-autonym="Español" data-language-local-name="španjolski" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Pol%C3%BCnoom" title="Polünoom – estonski" lang="et" hreflang="et" data-title="Polünoom" data-language-autonym="Eesti" data-language-local-name="estonski" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Polinomio" title="Polinomio – baskijski" lang="eu" hreflang="eu" data-title="Polinomio" data-language-autonym="Euskara" data-language-local-name="baskijski" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%D9%86%D8%AF%D8%AC%D9%85%D9%84%D9%87%E2%80%8C%D8%A7%DB%8C" title="چندجمله‌ای – perzijski" lang="fa" hreflang="fa" data-title="چندجمله‌ای" data-language-autonym="فارسی" data-language-local-name="perzijski" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Polynomi" title="Polynomi – finski" lang="fi" hreflang="fi" data-title="Polynomi" data-language-autonym="Suomi" data-language-local-name="finski" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Polyn%C3%B4me" title="Polynôme – francuski" lang="fr" hreflang="fr" data-title="Polynôme" data-language-autonym="Français" data-language-local-name="francuski" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-fy mw-list-item"><a href="https://fy.wikipedia.org/wiki/Mearterm" title="Mearterm – zapadnofrizijski" lang="fy" hreflang="fy" data-title="Mearterm" data-language-autonym="Frysk" data-language-local-name="zapadnofrizijski" class="interlanguage-link-target"><span>Frysk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Ilt%C3%A9armach" title="Iltéarmach – irski" lang="ga" hreflang="ga" data-title="Iltéarmach" data-language-autonym="Gaeilge" data-language-local-name="irski" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Polinomio" title="Polinomio – galicijski" lang="gl" hreflang="gl" data-title="Polinomio" data-language-autonym="Galego" data-language-local-name="galicijski" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%95%D7%9C%D7%99%D7%A0%D7%95%D7%9D" title="פולינום – hebrejski" lang="he" hreflang="he" data-title="פולינום" data-language-autonym="עברית" data-language-local-name="hebrejski" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AA%E0%A4%A6" title="बहुपद – hindski" lang="hi" hreflang="hi" data-title="बहुपद" data-language-autonym="हिन्दी" data-language-local-name="hindski" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Polinom" title="Polinom – mađarski" lang="hu" hreflang="hu" data-title="Polinom" data-language-autonym="Magyar" data-language-local-name="mađarski" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A1%D5%A6%D5%B4%D5%A1%D5%B6%D5%A4%D5%A1%D5%B4" title="Բազմանդամ – armenski" lang="hy" hreflang="hy" data-title="Բազմանդամ" data-language-autonym="Հայերեն" data-language-local-name="armenski" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-iba mw-list-item"><a href="https://iba.wikipedia.org/wiki/Polinomial" title="Polinomial – iban" lang="iba" hreflang="iba" data-title="Polinomial" data-language-autonym="Jaku Iban" data-language-local-name="iban" class="interlanguage-link-target"><span>Jaku Iban</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Polinomial" title="Polinomial – indonezijski" lang="id" hreflang="id" data-title="Polinomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijski" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Polinomio" title="Polinomio – ido" lang="io" hreflang="io" data-title="Polinomio" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Margli%C3%B0a" title="Margliða – islandski" lang="is" hreflang="is" data-title="Margliða" data-language-autonym="Íslenska" data-language-local-name="islandski" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Polinomio" title="Polinomio – talijanski" lang="it" hreflang="it" data-title="Polinomio" data-language-autonym="Italiano" data-language-local-name="talijanski" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – japanski" lang="ja" hreflang="ja" data-title="多項式" data-language-autonym="日本語" data-language-local-name="japanski" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%A0%E1%83%90%E1%83%95%E1%83%90%E1%83%9A%E1%83%AC%E1%83%94%E1%83%95%E1%83%A0%E1%83%98" title="მრავალწევრი – gruzijski" lang="ka" hreflang="ka" data-title="მრავალწევრი" data-language-autonym="ქართული" data-language-local-name="gruzijski" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D3%A9%D0%BF%D0%BC%D2%AF%D1%88%D0%B5%D0%BB%D1%96%D0%BA" title="Көпмүшелік – kazaški" lang="kk" hreflang="kk" data-title="Көпмүшелік" data-language-autonym="Қазақша" data-language-local-name="kazaški" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-km mw-list-item"><a href="https://km.wikipedia.org/wiki/%E1%9E%96%E1%9E%A0%E1%9E%BB%E1%9E%92%E1%9E%B6" title="ពហុធា – kmerski" lang="km" hreflang="km" data-title="ពហុធា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="kmerski" class="interlanguage-link-target"><span>ភាសាខ្មែរ</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%A4%ED%95%AD%EC%8B%9D" title="다항식 – korejski" lang="ko" hreflang="ko" data-title="다항식" data-language-autonym="한국어" data-language-local-name="korejski" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Polynomium" title="Polynomium – latinski" lang="la" hreflang="la" data-title="Polynomium" data-language-autonym="Latina" data-language-local-name="latinski" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Polinomas" title="Polinomas – litavski" lang="lt" hreflang="lt" data-title="Polinomas" data-language-autonym="Lietuvių" data-language-local-name="litavski" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Polinoms" title="Polinoms – latvijski" lang="lv" hreflang="lv" data-title="Polinoms" data-language-autonym="Latviešu" data-language-local-name="latvijski" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mg mw-list-item"><a href="https://mg.wikipedia.org/wiki/Maromiantoana" title="Maromiantoana – malgaški" lang="mg" hreflang="mg" data-title="Maromiantoana" data-language-autonym="Malagasy" data-language-local-name="malgaški" class="interlanguage-link-target"><span>Malagasy</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – makedonski" lang="mk" hreflang="mk" data-title="Полином" data-language-autonym="Македонски" data-language-local-name="makedonski" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AC%E0%B4%B9%E0%B5%81%E0%B4%AA%E0%B4%A6%E0%B4%82" title="ബഹുപദം – malajalamski" lang="ml" hreflang="ml" data-title="ബഹുപദം" data-language-autonym="മലയാളം" data-language-local-name="malajalamski" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%AC%E0%A4%B9%E0%A5%81%E0%A4%AA%E0%A4%A6%E0%A5%80" title="बहुपदी – marathski" lang="mr" hreflang="mr" data-title="बहुपदी" data-language-autonym="मराठी" data-language-local-name="marathski" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Polinomial" title="Polinomial – malajski" lang="ms" hreflang="ms" data-title="Polinomial" data-language-autonym="Bahasa Melayu" data-language-local-name="malajski" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Polynoom" title="Polynoom – nizozemski" lang="nl" hreflang="nl" data-title="Polynoom" data-language-autonym="Nederlands" data-language-local-name="nizozemski" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Polynom" title="Polynom – norveški nynorsk" lang="nn" hreflang="nn" data-title="Polynom" data-language-autonym="Norsk nynorsk" data-language-local-name="norveški nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Polynom" title="Polynom – norveški bokmål" lang="nb" hreflang="nb" data-title="Polynom" data-language-autonym="Norsk bokmål" data-language-local-name="norveški bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wielomian" title="Wielomian – poljski" lang="pl" hreflang="pl" data-title="Wielomian" data-language-autonym="Polski" data-language-local-name="poljski" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt badge-Q70893996 mw-list-item" title=""><a href="https://pt.wikipedia.org/wiki/Polin%C3%B3mio" title="Polinómio – portugalski" lang="pt" hreflang="pt" data-title="Polinómio" data-language-autonym="Português" data-language-local-name="portugalski" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Polinom" title="Polinom – rumunjski" lang="ro" hreflang="ro" data-title="Polinom" data-language-autonym="Română" data-language-local-name="rumunjski" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – ruski" lang="ru" hreflang="ru" data-title="Многочлен" data-language-autonym="Русский" data-language-local-name="ruski" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Polinom" title="Polinom – srpsko-hrvatski" lang="sh" hreflang="sh" data-title="Polinom" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="srpsko-hrvatski" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B6%B6%E0%B7%84%E0%B7%94_%E0%B6%B4%E0%B6%AF%E0%B6%BA" title="බහු පදය – sinhaleški" lang="si" hreflang="si" data-title="බහු පදය" data-language-autonym="සිංහල" data-language-local-name="sinhaleški" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Polynomial" title="Polynomial – Simple English" lang="en-simple" hreflang="en-simple" data-title="Polynomial" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Mnoho%C4%8Dlen" title="Mnohočlen – slovački" lang="sk" hreflang="sk" data-title="Mnohočlen" data-language-autonym="Slovenčina" data-language-local-name="slovački" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Polinom" title="Polinom – slovenski" lang="sl" hreflang="sl" data-title="Polinom" data-language-autonym="Slovenščina" data-language-local-name="slovenski" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Polinomet" title="Polinomet – albanski" lang="sq" hreflang="sq" data-title="Polinomet" data-language-autonym="Shqip" data-language-local-name="albanski" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BB%D0%B8%D0%BD%D0%BE%D0%BC" title="Полином – srpski" lang="sr" hreflang="sr" data-title="Полином" data-language-autonym="Српски / srpski" data-language-local-name="srpski" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Polynom" title="Polynom – švedski" lang="sv" hreflang="sv" data-title="Polynom" data-language-autonym="Svenska" data-language-local-name="švedski" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B2%E0%AF%8D%E0%AE%B2%E0%AF%81%E0%AE%B1%E0%AF%81%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95%E0%AF%8B%E0%AE%B5%E0%AF%88" title="பல்லுறுப்புக்கோவை – tamilski" lang="ta" hreflang="ta" data-title="பல்லுறுப்புக்கோவை" data-language-autonym="தமிழ்" data-language-local-name="tamilski" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-tg mw-list-item"><a href="https://tg.wikipedia.org/wiki/%D0%91%D0%B8%D1%81%D1%91%D1%80%D1%8A%D1%83%D0%B7%D0%B2%D0%B0" title="Бисёръузва – tadžički" lang="tg" hreflang="tg" data-title="Бисёръузва" data-language-autonym="Тоҷикӣ" data-language-local-name="tadžički" class="interlanguage-link-target"><span>Тоҷикӣ</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%9E%E0%B8%AB%E0%B8%B8%E0%B8%99%E0%B8%B2%E0%B8%A1" title="พหุนาม – tajlandski" lang="th" hreflang="th" data-title="พหุนาม" data-language-autonym="ไทย" data-language-local-name="tajlandski" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Polynomial" title="Polynomial – tagalog" lang="tl" hreflang="tl" data-title="Polynomial" data-language-autonym="Tagalog" data-language-local-name="tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Polinom" title="Polinom – turski" lang="tr" hreflang="tr" data-title="Polinom" data-language-autonym="Türkçe" data-language-local-name="turski" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%BD%D0%BE%D0%B3%D0%BE%D1%87%D0%BB%D0%B5%D0%BD" title="Многочлен – ukrajinski" lang="uk" hreflang="uk" data-title="Многочлен" data-language-autonym="Українська" data-language-local-name="ukrajinski" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%DA%A9%D8%AB%DB%8C%D8%B1_%D8%A7%D9%84%D8%A7%D8%B3%D9%85%DB%8C" title="کثیر الاسمی – urdski" lang="ur" hreflang="ur" data-title="کثیر الاسمی" data-language-autonym="اردو" data-language-local-name="urdski" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Ko%CA%BBphad" title="Koʻphad – uzbečki" lang="uz" hreflang="uz" data-title="Koʻphad" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbečki" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90a_th%E1%BB%A9c" title="Đa thức – vijetnamski" lang="vi" hreflang="vi" data-title="Đa thức" data-language-autonym="Tiếng Việt" data-language-local-name="vijetnamski" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F" title="多项式 – wu kineski" lang="wuu" hreflang="wuu" data-title="多项式" data-language-autonym="吴语" data-language-local-name="wu kineski" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-yi mw-list-item"><a href="https://yi.wikipedia.org/wiki/%D7%A4%D7%90%D7%9C%D7%99%D7%A0%D7%90%D7%9D" title="פאלינאם – jidiš" lang="yi" hreflang="yi" data-title="פאלינאם" data-language-autonym="ייִדיש" data-language-local-name="jidiš" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="dobri članak"><a href="https://yo.wikipedia.org/wiki/On%C3%ADr%C3%BAiyep%C3%BAp%E1%BB%8D%CC%80" title="Onírúiyepúpọ̀ – jorupski" lang="yo" hreflang="yo" data-title="Onírúiyepúpọ̀" data-language-autonym="Yorùbá" data-language-local-name="jorupski" class="interlanguage-link-target"><span>Yorùbá</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – kineski" lang="zh" hreflang="zh" data-title="多項式" data-language-autonym="中文" data-language-local-name="kineski" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="多項式" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%A4%9A%E9%A0%85%E5%BC%8F" title="多項式 – kantonski" lang="yue" hreflang="yue" data-title="多項式" data-language-autonym="粵語" data-language-local-name="kantonski" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span 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href="/w/index.php?title=Polinom&printable=yes" title="Inačica za ispis ove stranice [p]" accesskey="p"><span>Inačica za ispis</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> Wikimedijini projekti </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/Polynomial" hreflang="en"><span>Zajednički poslužitelj</span></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/Q43260" title="Poveznica na stavku na projektu Wikipodatci [g]" accesskey="g"><span>Stavka na Wikipodatcima</span></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="Pomagala"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Izgled"> <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">Izgled</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">premjesti</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">sakrij</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">Izvor: Wikipedija</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="hr" dir="ltr"><p><b>Polinom</b> je <a href="/wiki/Matematika" title="Matematika">matematička</a> <a href="/wiki/Funkcija_(matematika)" title="Funkcija (matematika)">funkcija</a> s jednom ili više <a href="/wiki/Varijabla" title="Varijabla">varijabla</a> koja se može zapisati kao <i>linearna kombinacija</i> umnožaka njihovih <a href="/wiki/Potencija" class="mw-redirect" title="Potencija">potencija</a>, odnosno kao zbroj monoma sastavljenih od umnožaka <a href="/wiki/Konstanta" title="Konstanta">konstante</a> i kombinacija potencija svake od varijabla. Polinom <i>n</i>-tog stupnja u jednoj varijabli je funkcija<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/573d074df5002c0d2e9492aaf343b9957c55d9f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.473ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}}" /></span></dd></dl> <p>u kojoj su koeficijenti je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbf42ecda092975c9c69dae84e16182ba5fe2e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{1}}" /></span>,..., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" /></span> <a href="/wiki/Konstanta" title="Konstanta">konstante</a> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf50204372c4d2e24c37aace0035de48b877973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.709ex; height:2.676ex;" alt="{\displaystyle a_{n}\neq 0}" /></span>. Broj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/693ad9f934775838bd72406b41ada4a59785d7ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.284ex; height:2.009ex;" alt="{\displaystyle a_{0}}" /></span> zove se slobodni koeficijent, a broj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" /></span> vodeći koeficijent. </p><p>Polinomi se skraćeno zapisuju pomoću simbola za zbrajanje, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f695568eae8df4a96e805d6032b59758030c61d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.884ex; height:6.843ex;" alt="{\displaystyle P(x)=\sum _{i=0}^{n}a_{i}x^{i}}" /></span> .</dd></dl> <p>Ponekad se polinomom zove sam polinomni izraz sa zbrojem raznih potencija neke veličine ili izraza, pa se pripadna funkcija navodi kao polinomna funkcija. </p><p>Stupanj polinoma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f7012fa931884835ca6879d037845a3597ff95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.019ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{0}}" /></span> za <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cf50204372c4d2e24c37aace0035de48b877973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.709ex; height:2.676ex;" alt="{\displaystyle a_{n}\neq 0}" /></span> je broj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span>. Pišemo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{deg}}P=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>deg</mtext> </mrow> <mi>P</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{deg}}P=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48dba08ce813e1e66236ad7f8a94c0356cb0a693" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.726ex; height:2.509ex;" alt="{\displaystyle {\text{deg}}P=n}" /></span>. </p><p>Polinomi imaju ključnu ulogu u proučavanju <a href="/wiki/Algebarski_broj" title="Algebarski broj">algebarskih brojeva</a> te su česti u drugim granama <a href="/wiki/Znanost" title="Znanost">znanosti</a> poput <a href="/wiki/Fizika" title="Fizika">fizike</a> i <a href="/wiki/Ra%C4%8Dunarstvo" title="Računarstvo">računarstva</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Monomi,_binomi,_trinomi,_itd."><span id="Monomi.2C_binomi.2C_trinomi.2C_itd."></span>Monomi, binomi, trinomi, itd.</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=1" title="Uredi odlomak: Monomi, binomi, trinomi, itd." class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=1" title="Uredi kôd odjeljka Monomi, binomi, trinomi, itd."><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pribrojnici u polinomu nazivaju se <b>monomi</b>; oni su i sami polinomi s jednim članom. Monom je umnožak konstante i bilo koje kombinacije potencija varijabla. Tako su, na primjer </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -7x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -7x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6624786b82d1160b377cea3c2b077839a7de8be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.355ex; height:2.843ex;" alt="{\displaystyle -7x^{2}}" /></span>,  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x^{2}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x^{2}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3af41c6067e57dd76d4c88ee1466d44962b77cb8" 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 3x^{2}y}" /></span>,  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{4}}xy^{5}z^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mstyle> </mrow> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{4}}xy^{5}z^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33f58d97e46da986a69c186ad388a982fae188e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.155ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{4}}xy^{5}z^{2}}" /></span>,  <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi y^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ξ</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi y^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83b75c90e3453375f220ccd54df249fd93ddd050" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.245ex; height:3.009ex;" alt="{\displaystyle \xi y^{4}}" /></span></dd></dl> <p>monomi u varijablama <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span>. </p><p>Polinom koji u temeljnom obliku ima samo dva člana naziva se <b>binom</b>. Polinom s tri člana je <b>trinom</b>. Tako je npr. kvadrat binoma jednak trinomu u dvije varijable: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,y)=(\underbrace {x-y} _{\text{binom}})^{2}=\underbrace {x^{2}-2xy+y^{2}} _{\text{trinom}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>binom</mtext> </mrow> </munder> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>trinom</mtext> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=(\underbrace {x-y} _{\text{binom}})^{2}=\underbrace {x^{2}-2xy+y^{2}} _{\text{trinom}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a779304b9bf58ffebfff267e10d156645cea0554" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-right: -0.028ex; width:35.415ex; height:6.509ex;" alt="{\displaystyle P(x,y)=(\underbrace {x-y} _{\text{binom}})^{2}=\underbrace {x^{2}-2xy+y^{2}} _{\text{trinom}}}" /></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x,y)=(\underbrace {x^{2}+y} _{\text{binom}})^{2}=\underbrace {x^{4}+2x^{2}y+y^{2}} _{\text{trinom}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>y</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>binom</mtext> </mrow> </munder> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>trinom</mtext> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=(\underbrace {x^{2}+y} _{\text{binom}})^{2}=\underbrace {x^{4}+2x^{2}y+y^{2}} _{\text{trinom}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f6bf02c1b5096655f5d146f4684a7ce20fdbe0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; margin-right: -0.028ex; width:37.523ex; height:6.509ex;" alt="{\displaystyle P(x,y)=(\underbrace {x^{2}+y} _{\text{binom}})^{2}=\underbrace {x^{4}+2x^{2}y+y^{2}} _{\text{trinom}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Nul-polinom">Nul-polinom</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=2" title="Uredi odlomak: Nul-polinom" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=2" title="Uredi kôd odjeljka Nul-polinom"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ako je polinom jednak nuli za sve vrijednosti svojih varijabli nazivamo ga <i>nul-polinom</i><sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> i za nj ne definiramo stupanj (ili se ponegdje formalno uzima da je njegov stupanj <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" /></span> ili <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac" 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 -1}" /></span>, ovisno o autoru). </p> <div class="mw-heading mw-heading2"><h2 id="Računske_operacije_s_polinomima"><span id="Ra.C4.8Dunske_operacije_s_polinomima"></span>Računske operacije s polinomima</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=3" title="Uredi odlomak: Računske operacije s polinomima" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=3" title="Uredi kôd odjeljka Računske operacije s polinomima"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dva polinoma možemo zbrajati, oduzimati, množiti i dijeliti. Zbrajanje i množenje je komutativno te vrijede uobičajena <a href="/wiki/Algebra" title="Algebra">algebarska</a> pravila. Rezultat dijeljenja dva polinoma nije uvijek polinom: očigledan primjer je dijeljenje polinoma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span>-tog stupnja polinomom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span>-tog stupnja kada je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n<m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo><</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n<m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cff096773597d7223f9d90162eb2d780dfc18dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle n<m}" /></span>. </p><p>Uočimo da oduzeti dva polinoma možemo tako da polinom koji je u funkciji umanjitelja pomnožimo s <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/311a5a9dcc81d10b9313cd9e32e254200165f2b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.78ex; height:2.843ex;" alt="{\displaystyle (-1)}" /></span> te ga zbrojimo s polinomom umanjenikom. </p> <div class="mw-heading mw-heading3"><h3 id="Primjeri">Primjeri</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=4" title="Uredi odlomak: Primjeri" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=4" title="Uredi kôd odjeljka Primjeri"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Uzmimo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=x^{2}+x-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{2}+x-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/becd5a20f42056c3b1159d8e3a93359794af1a20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.54ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{2}+x-2}" /></span> i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1460d1a0363d3d2e7b8142d3d2a9792edb1d6027" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.408ex; height:2.843ex;" alt="{\displaystyle Q(x)=x-1}" /></span>. </p><p>Njihovi zbroj i umnožak su: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)+Q(x)=x^{2}+2x-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)+Q(x)=x^{2}+2x-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a3a69328cb1833104049ba2cc7aadc254d428b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.52ex; height:3.176ex;" alt="{\displaystyle P(x)+Q(x)=x^{2}+2x-3}" /></span>,</dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)\cdot Q(x)=x^{3}-3x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)\cdot Q(x)=x^{3}-3x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a916200cc6401ccba3153799856479d53b95fef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.359ex; height:3.176ex;" alt="{\displaystyle P(x)\cdot Q(x)=x^{3}-3x+2}" /></span>.</dd></dl> <p>Opišimo kako algoritamski podijeliti ova dva polinoma. Ovdje je, radi jednostavnosti, rezultat dijeljenja polinom. Želimo izračunati <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P(x)}{Q(x)}}=R(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {P(x)}{Q(x)}}=R(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d3864affb252792adac4cc739a48ad5cf83e8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.462ex; height:6.509ex;" alt="{\displaystyle {\frac {P(x)}{Q(x)}}=R(x).}" /></span> </p><p>Ta jednadžba ekvivalentna je s <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+x-2=(x-1)R(x).}"> <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> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+x-2=(x-1)R(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01c0658e75f1019c1a7062d277bf4fbf19ee4736" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.347ex; height:3.176ex;" alt="{\displaystyle x^{2}+x-2=(x-1)R(x).}" /></span> Prvi član polinoma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5e851b43895fbe06436240dc7daa4d2033f082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.903ex; height:2.843ex;" alt="{\displaystyle R(x)}" /></span> jednak je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span> jer množenjem s <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c34249c90218860e305146c894d2c0a0b2b662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.142ex; height:2.843ex;" alt="{\displaystyle (x-1)}" /></span> mora dati član s jediničnim koeficijentom najveće potencije 2: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+\dots )}"> <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>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mi>x</mi> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87fd2bc0ba49da5cab146a989b2e17e64ccec0a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.309ex; height:3.176ex;" alt="{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+\dots )}" /></span> </p><p>Ostatak (...) je neki polinom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}" /></span> pa u prvom koraku imamo </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+T(x))}"> <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>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mi>x</mi> </mstyle> </mrow> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+T(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5c94be96e2527cbff430e745b3d23729845baba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.361ex; height:3.176ex;" alt="{\displaystyle (x^{2}+x-2)=(x-1)({\color {Red}x}+T(x))}" /></span></dd></dl> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{2}+x-2)=x^{2}-x+(x-1)T(x)}"> <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>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x^{2}+x-2)=x^{2}-x+(x-1)T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55b6eb158bdfaf6b6e751e74aebecbc5a2dc3727" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.776ex; height:3.176ex;" alt="{\displaystyle (x^{2}+x-2)=x^{2}-x+(x-1)T(x)}" /></span>.</dd></dl> <p>Dobivamo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x-2=(x-1)T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x-2=(x-1)T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9135641bdfedec08261eb4a7170f3433e89b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.511ex; height:2.843ex;" alt="{\displaystyle 2x-2=(x-1)T(x)}" /></span> čime je problem dijeljenja sveden na dijeljenje polinoma stupnja nižeg za 1. </p><p>U drugom koraku rješavamo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x-2=(x-1)T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x-2=(x-1)T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9135641bdfedec08261eb4a7170f3433e89b124" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.511ex; height:2.843ex;" alt="{\displaystyle 2x-2=(x-1)T(x)}" /></span>. </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1171c29b4c2b5575f50a4ea9313f90448a2cbe05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.775ex; height:2.843ex;" alt="{\displaystyle T(x)}" /></span> može jedino biti polinom stupnja 0 jer množeći <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94c34249c90218860e305146c894d2c0a0b2b662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.142ex; height:2.843ex;" alt="{\displaystyle (x-1)}" /></span> ne smije dati potencije veće od 1: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x-2=(x-1)({\color {Red}2}+\dots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>2</mn> </mstyle> </mrow> <mo>+</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x-2=(x-1)({\color {Red}2}+\dots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05795d27ad3b5913a635a775cf8092d7ae73ba66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.271ex; height:2.843ex;" alt="{\displaystyle 2x-2=(x-1)({\color {Red}2}+\dots )}" /></span>. </p><p>Ostatak (...) može biti samo nulpolinom, tj. 0. Mogli smo uočiti da je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x-2=(x-1)\cdot 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x-2=(x-1)\cdot 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14a92f7390253f1d4f2086964f3c068444ed751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.577ex; height:2.843ex;" alt="{\displaystyle 2x-2=(x-1)\cdot 2}" /></span>. </p><p>Rješenje je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=({\color {Red}x}+{\color {Red}2}+{\color {Red}0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mi>x</mi> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>2</mn> </mstyle> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#ED1B23"> <mn>0</mn> </mstyle> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=({\color {Red}x}+{\color {Red}2}+{\color {Red}0})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ef27e8d00d3a7d70c88b9900ee3601eaa9b0e06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.146ex; height:2.843ex;" alt="{\displaystyle R(x)=({\color {Red}x}+{\color {Red}2}+{\color {Red}0})}" /></span>, odnosno <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(x)=x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(x)=x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22d3379d56365b4e119e5c8244b54e93e97c496c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.334ex; height:2.843ex;" alt="{\displaystyle R(x)=x+2}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Uporaba_polinoma">Uporaba polinoma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=5" title="Uredi odlomak: Uporaba polinoma" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=5" title="Uredi kôd odjeljka Uporaba polinoma"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div><p> Zbog jednostavnosti računanja s polinomima, posebno njihovog strojnog izvrjednjavanja, vrijedosti mnogih drugih funkcija često se aproksimiraju polinomom određenog stupnja na određenom intervalu. Ako je vrijednost funkcije poznata u konačno mnogo točaka, vrijednosti između točaka mogu se procijeniti <a href="/wiki/Interpolacija" title="Interpolacija">interpolacijom</a> iz polinoma koji u tim točkama daje egzaktne vrijednosti<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> ili <a href="/w/index.php?title=Regresija&action=edit&redlink=1" class="new" title="Regresija (stranica ne postoji)">regresijom</a> uz pomoć polinoma po volji izabranog stupnja koji po svim poznatim točkama daje najmanju pogrešku. </p><ul class="gallery mw-gallery-traditional"> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datoteka:Interpolation_Data.svg" class="mw-file-description" title="Vrijednosti nepoznate funkcije poznate su u sedam točaka"><img alt="Vrijednosti nepoznate funkcije poznate su u sedam točaka" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Interpolation_Data.svg/120px-Interpolation_Data.svg.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Interpolation_Data.svg/180px-Interpolation_Data.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Interpolation_Data.svg/240px-Interpolation_Data.svg.png 2x" data-file-width="600" data-file-height="480" /></a></span></div> <div class="gallerytext">Vrijednosti nepoznate funkcije poznate su u sedam točaka</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datoteka:Interpolation_example_linear.svg" class="mw-file-description" title="Vrijednosti između točaka mogu se procijeniti linearnom interpolacijom od točke do točke"><img alt="Vrijednosti između točaka mogu se procijeniti linearnom interpolacijom od točke do točke" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Interpolation_example_linear.svg/120px-Interpolation_example_linear.svg.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Interpolation_example_linear.svg/180px-Interpolation_example_linear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Interpolation_example_linear.svg/240px-Interpolation_example_linear.svg.png 2x" data-file-width="600" data-file-height="480" /></a></span></div> <div class="gallerytext">Vrijednosti između točaka mogu se procijeniti linearnom interpolacijom od točke do točke</div> </li> <li class="gallerybox" style="width: 155px"> <div class="thumb" style="width: 150px; height: 150px;"><span typeof="mw:File"><a href="/wiki/Datoteka:Interpolation_example_polynomial.svg" class="mw-file-description" title="Vrijednosti između točaka procijenjene interpolacijskim polinomom šestoga stupnja"><img alt="Vrijednosti između točaka procijenjene interpolacijskim polinomom šestoga stupnja" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Interpolation_example_polynomial.svg/120px-Interpolation_example_polynomial.svg.png" decoding="async" width="120" height="96" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/41/Interpolation_example_polynomial.svg/180px-Interpolation_example_polynomial.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/41/Interpolation_example_polynomial.svg/240px-Interpolation_example_polynomial.svg.png 2x" data-file-width="600" data-file-height="480" /></a></span></div> <div class="gallerytext">Vrijednosti između točaka procijenjene interpolacijskim polinomom šestoga stupnja</div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Nultočke_polinoma"><span id="Nulto.C4.8Dke_polinoma"></span>Nultočke polinoma</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=6" title="Uredi odlomak: Nultočke polinoma" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=6" title="Uredi kôd odjeljka Nultočke polinoma"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>U primjeni, kao i u teoriji, često je potrebno znati u kojim točkama polinomi poprimaju vrijednost nula. Te se točke nazivaju <a href="/wiki/Nulto%C4%8Dka" title="Nultočka">nultočkama</a> ili korijenima polinoma. Ako je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span> nultočka polinoma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" /></span>, vrijedi <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(\alpha )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(\alpha )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c47730d546c54586d0538f7d6ed04a00dd6f025d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.303ex; height:2.843ex;" alt="{\displaystyle P(\alpha )=0}" /></span>. Prema <a href="/w/index.php?title=B%C3%A9zoutov_pou%C4%8Dak_za_polinome&action=edit&redlink=1" class="new" title="Bézoutov poučak za polinome (stranica ne postoji)">Bézoutovom poučku za polinome</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7452ea0e1fdee917d5766a3dbb9ebbfe46f4590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.467ex; height:2.843ex;" alt="{\displaystyle (x-\alpha )}" /></span> tada dijeli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" /></span>.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Iz ovog poučka i <a href="/w/index.php?title=Osnovni_teorem_algebre&action=edit&redlink=1" class="new" title="Osnovni teorem algebre (stranica ne postoji)">osnovnog teorema algebre</a>, koji kaže da svaki polinom stupnja većeg od nule ima nultočku u skupu <a href="/wiki/Kompleksni_broj" title="Kompleksni broj">kompleksnih brojeva</a>, slijedi da svaki polinom <i>n</i>-tog stupnja u jednoj varijabli ima točno <i>n</i> nultočaka u skupu kompleksnih brojeva, s tim da pritom neke nultočke mogu biti višestruke kratnosti, odnosno da za neke nultočke može i <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-\alpha )^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-\alpha )^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d896eda63db8e7f883ab00b750d8a75e81c9c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.556ex; height:3.176ex;" alt="{\displaystyle (x-\alpha )^{k}}" /></span> dijeliti <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" /></span>, gdje se najveći takav <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}" /></span> naziva kratnošću nultočke. Vrijedi i sljedeće: ako su <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d69430689b8aac2832684109cde587c4ae828d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{1}}" /></span>,<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aef5f08a56f51deb324e5eed2fb9b2e3c279889d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.542ex; height:2.009ex;" alt="{\displaystyle \alpha _{2}}" /></span>, ..., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87bc96d36d1f4d3c81e942c864b212b2ab50b45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.706ex; height:2.009ex;" alt="{\displaystyle \alpha _{n}}" /></span> kompleksne nultočke polinoma <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}" /></span> s vodećim koeficijentom <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}" /></span>, on se može na jedinstven način zapisati kao umnožak <i>n</i> polinoma prvoga stupnja,<sup id="cite_ref-:0_1-3" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{n}(x-\alpha _{1})(x-\alpha _{2})\dots (x-\alpha _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{n}(x-\alpha _{1})(x-\alpha _{2})\dots (x-\alpha _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b25a9a07db49c66e10fcd8c98d9666b81cb3100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.657ex; height:2.843ex;" alt="{\displaystyle P(x)=a_{n}(x-\alpha _{1})(x-\alpha _{2})\dots (x-\alpha _{n})}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Izvori">Izvori</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinom&veaction=edit&section=7" title="Uredi odlomak: Izvori" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinom&action=edit&section=7" title="Uredi kôd odjeljka Izvori"><span>uredi kôd</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r6541845">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-:0_1-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r7012897">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"»""«"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}</style><cite id="CITEREFBujanovićMuha2018" class="citation book cs1">Bujanović, Zvonimir; Muha, Boris. 2018. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20191219215513/https://web.math.pmf.unizg.hr/nastava/em/EM1/materijali/em1-skripta.pdf"><i>Elementarna matematika I</i></a> <span class="cs1-format">(PDF)</span>. Prirodoslovno-matematički fakultet. Zagreb. Inačica izvorne stranice arhivirana 19. prosinca 2019<span class="reference-accessdate">. Pristupljeno 8. travnja 2021.</span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementarna+matematika+I&rft.place=Zagreb&rft.pub=Prirodoslovno-matemati%C4%8Dki+fakultet&rft.date=2018&rft.aulast=Bujanovi%C4%87&rft.aufirst=Zvonimir&rft.au=Muha%2C+Boris&rft_id=https%3A%2F%2Fweb.math.pmf.unizg.hr%2Fnastava%2Fem%2FEM1%2Fmaterijali%2Fem1-skripta.pdf&rfr_id=info%3Asid%2Fhr.wikipedia.org%3APolinom" class="Z3988"></span><span class="cs1-maint citation-comment">CS1 održavanje: bot: nepoznat status originalnog URL-a (<a href="/wiki/Kategorija:CS1_odr%C5%BEavanje:_bot:_nepoznat_status_originalnog_URL-a" title="Kategorija:CS1 održavanje: bot: nepoznat status originalnog URL-a">link</a>)</span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r7012897" /><cite id="CITEREFPavković1990" class="citation book cs1">Pavković, Boris. 1990. <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/456589919"><i>Polinomi</i></a>. 4. izd izdanje. Školska knjiga. Zagreb. <a href="/wiki/ISBN" class="mw-redirect" title="ISBN">ISBN</a> <a href="/wiki/Posebno:Tra%C5%BEi_ISBN/86-03-99890-6" title="Posebno:Traži ISBN/86-03-99890-6"><bdi>86-03-99890-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Polinomi&rft.place=Zagreb&rft.edition=4.+izd&rft.pub=%C5%A0kolska+knjiga&rft.date=1990&rft.isbn=86-03-99890-6&rft.aulast=Pavkovi%C4%87&rft.aufirst=Boris&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F456589919&rfr_id=info%3Asid%2Fhr.wikipedia.org%3APolinom" class="Z3988"></span></span> </li> </ol></div></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Dobavljeno iz "<a dir="ltr" href="https://hr.wikipedia.org/w/index.php?title=Polinom&oldid=6434676">https://hr.wikipedia.org/w/index.php?title=Polinom&oldid=6434676</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Posebno:Kategorije" title="Posebno:Kategorije">Kategorija</a>: <ul><li><a href="/wiki/Kategorija:Algebra" title="Kategorija:Algebra">Algebra</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Skrivena kategorija: <ul><li><a href="/wiki/Kategorija:CS1_odr%C5%BEavanje:_bot:_nepoznat_status_originalnog_URL-a" title="Kategorija:CS1 održavanje: bot: nepoznat status originalnog URL-a">CS1 održavanje: bot: nepoznat status originalnog URL-a</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"> Ova stranica posljednji je put uređivana 30. srpnja 2022. u 15:47.</li> <li id="footer-info-copyright">Tekst je dostupan pod licencijom <a rel="nofollow" class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.hr">Creative Commons: Imenuj autora/Dijeli pod istim uvjetima</a>; mogu se primjenjivati i dodatni uvjeti. 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