Polinomio - Wikipedia, entziklopedia askea.

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ez da ezinbestekoa, ordea." class=""><span>Sortu kontua</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Berezi:SaioaHasi&returnto=Polinomio" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o" class=""><span>Hasi saioa</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Aukera gehiago" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Tresna pertsonalak" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Tresna pertsonalak</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_eu.wikipedia.org&uselang=eu"><span>Dohaintza egin</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:KontuaSortu&returnto=Polinomio" title="Kontu bat sortu eta horrekin saioa hastea eskatu nahi genizuke; ez da ezinbestekoa, ordea."><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Sortu kontua</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Berezi:SaioaHasi&returnto=Polinomio" title="Izen ematera gonbidatzen zaitugu. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Hasi saioa</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Izena eman gabeko erabiltzaileentzako orrialdeak <a href="/wiki/Laguntza:Sarrera" aria-label="Artikuluak aldatzeari buruz gehiago ikasi"><span>gehiago ikasi</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Berezi:NireEkarpenak" title="IP helbide honetatik egindako aldaketen zerrenda [y]" accesskey="y"><span>Ekarpenak</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Berezi:NireEztabaida" title="Zure IParen eztabaida [n]" accesskey="n"><span>Eztabaida</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Gunea"> <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="Edukiak" 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">Edukiak</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">mugitu alboko barrara</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">ezkutatu</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">⇑ Gora</div> </a> </li> <li id="toc-Historia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Historia</span> </div> </a> <ul id="toc-Historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monomioa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Monomioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Monomioa</span> </div> </a> <ul id="toc-Monomioa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aplikazioak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Aplikazioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Aplikazioak</span> </div> </a> <ul id="toc-Aplikazioak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definizio_aljebraikoa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definizio_aljebraikoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Definizio aljebraikoa</span> </div> </a> <button aria-controls="toc-Definizio_aljebraikoa-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>Erakutsi/ezkutatu Definizio aljebraikoa azpiatal</span> </button> <ul id="toc-Definizio_aljebraikoa-sublist" class="vector-toc-list"> <li id="toc-Aldagai_bakarreko_polinomioak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aldagai_bakarreko_polinomioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Aldagai bakarreko polinomioak</span> </div> </a> <ul id="toc-Aldagai_bakarreko_polinomioak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aldagai_anitzeko_polinomioak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Aldagai_anitzeko_polinomioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Aldagai anitzeko polinomioak</span> </div> </a> <ul id="toc-Aldagai_anitzeko_polinomioak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polinomioaren_maila" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polinomioaren_maila"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Polinomioaren maila</span> </div> </a> <ul id="toc-Polinomioaren_maila-sublist" class="vector-toc-list"> <li id="toc-Notazioa" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Notazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Notazioa</span> </div> </a> <ul id="toc-Notazioa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Monomioak" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Monomioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>Monomioak</span> </div> </a> <ul id="toc-Monomioak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polinomioak" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Polinomioak"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.3</span> <span>Polinomioak</span> </div> </a> <ul id="toc-Polinomioak-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Eragiketak_polinomioekin" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Eragiketak_polinomioekin"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Eragiketak polinomioekin</span> </div> </a> <button aria-controls="toc-Eragiketak_polinomioekin-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>Erakutsi/ezkutatu Eragiketak polinomioekin azpiatal</span> </button> <ul id="toc-Eragiketak_polinomioekin-sublist" class="vector-toc-list"> <li id="toc-Polinomioa_puntu_batean_ebaluatzea" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polinomioa_puntu_batean_ebaluatzea"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Polinomioa puntu batean ebaluatzea</span> </div> </a> <ul id="toc-Polinomioa_puntu_batean_ebaluatzea-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Batuketa_eta_kenketa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Batuketa_eta_kenketa"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Batuketa eta kenketa</span> </div> </a> <ul id="toc-Batuketa_eta_kenketa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Biderketa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Biderketa"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Biderketa</span> </div> </a> <ul id="toc-Biderketa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Identitate_nabarmenak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Identitate_nabarmenak"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Identitate nabarmenak</span> </div> </a> <ul id="toc-Identitate_nabarmenak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zatiketa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zatiketa"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Zatiketa</span> </div> </a> <ul id="toc-Zatiketa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Ariketak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ariketak"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span><figcaption></figcaption><p> Ariketak</p></span> </div> </a> <button aria-controls="toc-Ariketak-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>Erakutsi/ezkutatu <figcaption></figcaption><p> Ariketak</p> azpiatal</span> </button> <ul id="toc-Ariketak-sublist" class="vector-toc-list"> <li id="toc-Ruffiniren_erregela" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ruffiniren_erregela"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Ruffiniren erregela</span> </div> </a> <ul id="toc-Ruffiniren_erregela-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Faktore_komuna_ateratzea" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Faktore_komuna_ateratzea"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Faktore komuna ateratzea</span> </div> </a> <ul id="toc-Faktore_komuna_ateratzea-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Polinomioen_erroak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polinomioen_erroak"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Polinomioen erroak</span> </div> </a> <ul id="toc-Polinomioen_erroak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polinomioen_faktorizazioa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polinomioen_faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Polinomioen faktorizazioa</span> </div> </a> <button aria-controls="toc-Polinomioen_faktorizazioa-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>Erakutsi/ezkutatu Polinomioen faktorizazioa azpiatal</span> </button> <ul id="toc-Polinomioen_faktorizazioa-sublist" class="vector-toc-list"> <li id="toc-Kronecker_metodoa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kronecker_metodoa"> <div class="vector-toc-text"> <span class="vector-toc-numb">8.1</span> <span>Kronecker metodoa</span> </div> </a> <ul id="toc-Kronecker_metodoa-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Erreferentziak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Erreferentziak"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Erreferentziak</span> </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kanpo_estekak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kanpo_estekak"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Kanpo estekak</span> </div> </a> <ul id="toc-Kanpo_estekak-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="Edukiak" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Eduki taularen ikusgarritasuna aldatu" > <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">Eduki taularen ikusgarritasuna aldatu</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">Polinomio</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="Joan beste hizkuntza batean idatzitako artikulu batera. 82 hizkuntzatan eskuragarri." > <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 hizkuntza</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 – afrikaansa" lang="af" hreflang="af" data-title="Polinoom" data-language-autonym="Afrikaans" data-language-local-name="afrikaansa" 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 – arabiera" lang="ar" hreflang="ar" data-title="متعددة الحدود" data-language-autonym="العربية" data-language-local-name="arabiera" 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 – asturiera" lang="ast" hreflang="ast" data-title="Polinomiu" data-language-autonym="Asturianu" data-language-local-name="asturiera" 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 – azerbaijanera" lang="az" hreflang="az" data-title="Çoxhədli" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" 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 – baxkirera" lang="ba" hreflang="ba" data-title="Күпбыуын" data-language-autonym="Башҡортса" data-language-local-name="baxkirera" 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 – bielorrusiera" lang="be" hreflang="be" data-title="Мнагачлен" data-language-autonym="Беларуская" data-language-local-name="bielorrusiera" 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 – bulgariera" lang="bg" hreflang="bg" data-title="Многочлен" data-language-autonym="Български" data-language-local-name="bulgariera" 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 – bengalera" lang="bn" hreflang="bn" data-title="বহুপদী" data-language-autonym="বাংলা" data-language-local-name="bengalera" 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 – bosniera" lang="bs" hreflang="bs" data-title="Polinom" data-language-autonym="Bosanski" data-language-local-name="bosniera" 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 – katalana" lang="ca" hreflang="ca" data-title="Polinomi" data-language-autonym="Català" data-language-local-name="katalana" 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 – erdialdeko kurduera" lang="ckb" hreflang="ckb" data-title="ڕادەدار" data-language-autonym="کوردی" data-language-local-name="erdialdeko kurduera" 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 – txekiera" lang="cs" hreflang="cs" data-title="Polynom" data-language-autonym="Čeština" data-language-local-name="txekiera" 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 – txuvaxera" lang="cv" hreflang="cv" data-title="Полином" data-language-autonym="Чӑвашла" data-language-local-name="txuvaxera" 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 – galesa" lang="cy" hreflang="cy" data-title="Polynomial" data-language-autonym="Cymraeg" data-language-local-name="galesa" 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 – daniera" lang="da" hreflang="da" data-title="Polynomium" data-language-autonym="Dansk" data-language-local-name="daniera" 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 – alemana" lang="de" hreflang="de" data-title="Polynom" data-language-autonym="Deutsch" data-language-local-name="alemana" 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 – Zazaki" lang="diq" hreflang="diq" data-title="Polinom" data-language-autonym="Zazaki" data-language-local-name="Zazaki" 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 – greziera" lang="el" hreflang="el" data-title="Πολυώνυμο" data-language-autonym="Ελληνικά" data-language-local-name="greziera" 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 – ingelesa" lang="en" hreflang="en" data-title="Polynomial" data-language-autonym="English" data-language-local-name="ingelesa" 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 – esperantoa" lang="eo" hreflang="eo" data-title="Polinomo" data-language-autonym="Esperanto" data-language-local-name="esperantoa" 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 – gaztelania" lang="es" hreflang="es" data-title="Polinomio" data-language-autonym="Español" data-language-local-name="gaztelania" 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 – estoniera" lang="et" hreflang="et" data-title="Polünoom" data-language-autonym="Eesti" data-language-local-name="estoniera" class="interlanguage-link-target"><span>Eesti</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 – persiera" lang="fa" hreflang="fa" data-title="چندجمله‌ای" data-language-autonym="فارسی" data-language-local-name="persiera" 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 – finlandiera" lang="fi" hreflang="fi" data-title="Polynomi" data-language-autonym="Suomi" data-language-local-name="finlandiera" 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 – frantsesa" lang="fr" hreflang="fr" data-title="Polynôme" data-language-autonym="Français" data-language-local-name="frantsesa" 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 – mendebaldeko frisiera" lang="fy" hreflang="fy" data-title="Mearterm" data-language-autonym="Frysk" data-language-local-name="mendebaldeko frisiera" 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 – irlandera" lang="ga" hreflang="ga" data-title="Iltéarmach" data-language-autonym="Gaeilge" data-language-local-name="irlandera" 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 – galiziera" lang="gl" hreflang="gl" data-title="Polinomio" data-language-autonym="Galego" data-language-local-name="galiziera" 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 – hebreera" lang="he" hreflang="he" data-title="פולינום" data-language-autonym="עברית" data-language-local-name="hebreera" 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 – hindia" lang="hi" hreflang="hi" data-title="बहुपद" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Polinom" title="Polinom – kroaziera" lang="hr" hreflang="hr" data-title="Polinom" data-language-autonym="Hrvatski" data-language-local-name="kroaziera" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Polinom" title="Polinom – hungariera" lang="hu" hreflang="hu" data-title="Polinom" data-language-autonym="Magyar" data-language-local-name="hungariera" 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 – armeniera" lang="hy" hreflang="hy" data-title="Բազմանդամ" data-language-autonym="Հայերեն" data-language-local-name="armeniera" 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 – ibanera" lang="iba" hreflang="iba" data-title="Polinomial" data-language-autonym="Jaku Iban" data-language-local-name="ibanera" 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 – indonesiera" lang="id" hreflang="id" data-title="Polinomial" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonesiera" 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 – idoa" lang="io" hreflang="io" data-title="Polinomio" data-language-autonym="Ido" data-language-local-name="idoa" 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 – islandiera" lang="is" hreflang="is" data-title="Margliða" data-language-autonym="Íslenska" data-language-local-name="islandiera" 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 – italiera" lang="it" hreflang="it" data-title="Polinomio" data-language-autonym="Italiano" data-language-local-name="italiera" 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 – japoniera" lang="ja" hreflang="ja" data-title="多項式" data-language-autonym="日本語" data-language-local-name="japoniera" 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 – georgiera" lang="ka" hreflang="ka" data-title="მრავალწევრი" data-language-autonym="ქართული" data-language-local-name="georgiera" 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 – kazakhera" lang="kk" hreflang="kk" data-title="Көпмүшелік" data-language-autonym="Қазақша" data-language-local-name="kazakhera" 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 – khemerera" lang="km" hreflang="km" data-title="ពហុធា" data-language-autonym="ភាសាខ្មែរ" data-language-local-name="khemerera" 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 – koreera" lang="ko" hreflang="ko" data-title="다항식" data-language-autonym="한국어" data-language-local-name="koreera" 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 – latina" lang="la" hreflang="la" data-title="Polynomium" data-language-autonym="Latina" data-language-local-name="latina" 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 – lituaniera" lang="lt" hreflang="lt" data-title="Polinomas" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" 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 – letoniera" lang="lv" hreflang="lv" data-title="Polinoms" data-language-autonym="Latviešu" data-language-local-name="letoniera" 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 – malgaxea" lang="mg" hreflang="mg" data-title="Maromiantoana" data-language-autonym="Malagasy" data-language-local-name="malgaxea" 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 – mazedoniera" lang="mk" hreflang="mk" data-title="Полином" data-language-autonym="Македонски" data-language-local-name="mazedoniera" 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 – malabarera" lang="ml" hreflang="ml" data-title="ബഹുപദം" data-language-autonym="മലയാളം" data-language-local-name="malabarera" 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 – marathera" lang="mr" hreflang="mr" data-title="बहुपदी" data-language-autonym="मराठी" data-language-local-name="marathera" 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 – malaysiera" lang="ms" hreflang="ms" data-title="Polinomial" data-language-autonym="Bahasa Melayu" data-language-local-name="malaysiera" 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 – nederlandera" lang="nl" hreflang="nl" data-title="Polynoom" data-language-autonym="Nederlands" data-language-local-name="nederlandera" 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 – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Polynom" data-language-autonym="Norsk nynorsk" data-language-local-name="nynorsk (norvegiera)" 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 – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Polynom" data-language-autonym="Norsk bokmål" data-language-local-name="bokmål (norvegiera)" 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 – poloniera" lang="pl" hreflang="pl" data-title="Wielomian" data-language-autonym="Polski" data-language-local-name="poloniera" 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 – portugesa" lang="pt" hreflang="pt" data-title="Polinómio" data-language-autonym="Português" data-language-local-name="portugesa" 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 – errumaniera" lang="ro" hreflang="ro" data-title="Polinom" data-language-autonym="Română" data-language-local-name="errumaniera" 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 – errusiera" lang="ru" hreflang="ru" data-title="Многочлен" data-language-autonym="Русский" data-language-local-name="errusiera" 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 – serbokroaziera" lang="sh" hreflang="sh" data-title="Polinom" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="serbokroaziera" 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 – sinhala" lang="si" hreflang="si" data-title="බහු පදය" data-language-autonym="සිංහල" data-language-local-name="sinhala" 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 – eslovakiera" lang="sk" hreflang="sk" data-title="Mnohočlen" data-language-autonym="Slovenčina" data-language-local-name="eslovakiera" 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 – esloveniera" lang="sl" hreflang="sl" data-title="Polinom" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" 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 – albaniera" lang="sq" hreflang="sq" data-title="Polinomet" data-language-autonym="Shqip" data-language-local-name="albaniera" 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 – serbiera" lang="sr" hreflang="sr" data-title="Полином" data-language-autonym="Српски / srpski" data-language-local-name="serbiera" 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 – suediera" lang="sv" hreflang="sv" data-title="Polynom" data-language-autonym="Svenska" data-language-local-name="suediera" 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 – tamilera" lang="ta" hreflang="ta" data-title="பல்லுறுப்புக்கோவை" data-language-autonym="தமிழ்" data-language-local-name="tamilera" 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 – tajikera" lang="tg" hreflang="tg" data-title="Бисёръузва" data-language-autonym="Тоҷикӣ" data-language-local-name="tajikera" 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 – thailandiera" lang="th" hreflang="th" data-title="พหุนาม" data-language-autonym="ไทย" data-language-local-name="thailandiera" 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 – tagaloa" lang="tl" hreflang="tl" data-title="Polynomial" data-language-autonym="Tagalog" data-language-local-name="tagaloa" 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 – turkiera" lang="tr" hreflang="tr" data-title="Polinom" data-language-autonym="Türkçe" data-language-local-name="turkiera" 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 – ukrainera" lang="uk" hreflang="uk" data-title="Многочлен" data-language-autonym="Українська" data-language-local-name="ukrainera" 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 – urdua" lang="ur" hreflang="ur" data-title="کثیر الاسمی" data-language-autonym="اردو" data-language-local-name="urdua" 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 – uzbekera" lang="uz" hreflang="uz" data-title="Koʻphad" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbekera" 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 – vietnamera" lang="vi" hreflang="vi" data-title="Đa thức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamera" 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 txinera" lang="wuu" hreflang="wuu" data-title="多项式" data-language-autonym="吴语" data-language-local-name="wu txinera" 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 – yiddisha" lang="yi" hreflang="yi" data-title="פאלינאם" data-language-autonym="ייִדיש" data-language-local-name="yiddisha" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-yo badge-Q17437798 badge-goodarticle mw-list-item" title="artikulu onak"><a href="https://yo.wikipedia.org/wiki/On%C3%ADr%C3%BAiyep%C3%BAp%E1%BB%8D%CC%80" title="Onírúiyepúpọ̀ – jorubera" lang="yo" hreflang="yo" data-title="Onírúiyepúpọ̀" data-language-autonym="Yorùbá" data-language-local-name="jorubera" 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 – txinera" lang="zh" hreflang="zh" data-title="多項式" data-language-autonym="中文" data-language-local-name="txinera" 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 – kantonera" lang="yue" hreflang="yue" data-title="多項式" data-language-autonym="粵語" data-language-local-name="kantonera" class="interlanguage-link-target"><span>粵語</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q43260#sitelinks-wikipedia" title="Aldatu hizkuntzen arteko loturak" class="wbc-editpage">Aldatu loturak</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Izen-tarteak"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a 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alt="Artikulu hau "Kalitatezko 2.000 artikulu 12-16 urteko ikasleentzat" proiektuaren parte da" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/19px-Hezkuntza_Programa_12-16_ikonoa.png" decoding="async" width="19" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/28px-Hezkuntza_Programa_12-16_ikonoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ea/Hezkuntza_Programa_12-16_ikonoa.png/38px-Hezkuntza_Programa_12-16_ikonoa.png 2x" data-file-width="747" data-file-height="794" /></a></span></div></div> </div> <div id="siteSub" class="noprint">Wikipedia, Entziklopedia askea</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="eu" dir="ltr"><p class="mw-empty-elt"> </p><p><a href="/wiki/Matematika" title="Matematika">Matematikan</a>, <b>polinomioa</b> <a href="/wiki/Aldagai_(argipena)" class="mw-disambig" title="Aldagai (argipena)">aldagai</a> batez edo gehiagoz eta zenbait <a href="/wiki/Konstante" class="mw-disambig" title="Konstante">konstantez</a> osaturiko <a href="/wiki/Adierazpen_(matematika)" class="mw-redirect" title="Adierazpen (matematika)">adierazpen matematiko</a> mugatu bat da. Aldagaiak eta konstanteak <a href="/wiki/Batuketa" title="Batuketa">batuketaz</a>, <a href="/wiki/Kenketa" title="Kenketa">kenketaz</a> eta <a href="/wiki/Biderketa" title="Biderketa">biderketaz</a> elkartzen dira, eta aldagai <a href="/wiki/Berretzaile" class="mw-redirect" title="Berretzaile">berretzaileek</a> ez-negatibo eta <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">osoak</a> izan behar dute. </p><p>Definizioz, koefizienteak A multzoan dituzten polinomioak <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}X^{n}+a_{n-1}x^{n-1}+...+a_{1}X+a_{0}|a_{i}\in A}"> <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> <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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}X^{n}+a_{n-1}x^{n-1}+...+a_{1}X+a_{0}|a_{i}\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e14fb1b99ca835750c0fcc4666d0f6f2e22ab806" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.068ex; height:3.176ex;" alt="{\displaystyle a_{n}X^{n}+a_{n-1}x^{n-1}+...+a_{1}X+a_{0}|a_{i}\in A}"></span> erako adierazpena da. </p><p>Hori dela eta, polinomioen <a href="/wiki/Multzo" title="Multzo">multzoa</a> honela definitu daiteke: <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[x]={a_{0}+...+a_{n}x^{n}|a_{i}\in A,n\in \mathbb {N} \cup \{0\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∪</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[x]={a_{0}+...+a_{n}x^{n}|a_{i}\in A,n\in \mathbb {N} \cup \{0\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/367cb019b5a9b00c242064f4ddf8ab8114508dd7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.741ex; height:2.843ex;" alt="{\displaystyle A[x]={a_{0}+...+a_{n}x^{n}|a_{i}\in A,n\in \mathbb {N} \cup \{0\}}}"></span> </p><p>Adibidez, polinomioak honako hauek dira: </p> <ul><li><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}-4x+7\,}"> <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> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{2}-4x+7\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0780b8b7e83b25fad5976c86dd0a4388dcaad187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.089ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{2}-4x+7\,}"></span></li> <li><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)=xy-x^{2}\,}"> <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> <mi>x</mi> <mi>y</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=xy-x^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f27ad7e2e921043bdf6749694162681a5695d8b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.269ex; height:3.176ex;" alt="{\displaystyle P(x,y)=xy-x^{2}\,}"></span></li> <li><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^{5}+7\,}"> <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>5</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{5}+7\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/941f01b97757bef96244e4cc6f8102a823d76aba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.757ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{5}+7\,}"></span></li> <li><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)=7,45\,}"> <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> <mn>7</mn> <mo>,</mo> <mn>45</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=7,45\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f76721d513e452b1f4392a5e1e54cddbdf286759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.891ex; height:2.843ex;" alt="{\displaystyle P(x)=7,45\,}"></span></li> <li><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)=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> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c9d27bfca57c1625cf50b9f517adf3a2badd63d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.532ex; height:2.843ex;" alt="{\displaystyle P(x)=0\,}"></span></li> <li><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)={\frac {x}{9}}={\frac {1}{9}}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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>x</mi> <mn>9</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>9</mn> </mfrac> </mrow> <mi>x</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)={\frac {x}{9}}={\frac {1}{9}}x\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40f4b80c7815c28fd44214c88cfd7bdc09f884c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.963ex; height:5.176ex;" alt="{\displaystyle P(x)={\frac {x}{9}}={\frac {1}{9}}x\,}"></span></li></ul> <p>Beste hauek, ordea, ez dira polinomioak, berretzaile negatibo edo ez-osoak dituztelako: </p> <ul><li><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}-{\frac {4}{x}}+7=x^{2}-4x^{-1}+7\,}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>x</mi> </mfrac> </mrow> <mo>+</mo> <mn>7</mn> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mn>7</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{2}-{\frac {4}{x}}+7=x^{2}-4x^{-1}+7\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc89f1386faa0df7d60a13baa0ee6e93ddc3b92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.914ex; height:5.176ex;" alt="{\displaystyle P(x)=x^{2}-{\frac {4}{x}}+7=x^{2}-4x^{-1}+7\,}"></span></li> <li><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)=x^{2}-4xy^{4,5}\,}"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mo>,</mo> <mn>5</mn> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=x^{2}-4xy^{4,5}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a35514e4abd116dcfda6aa36965eccac787c425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.77ex; height:3.176ex;" alt="{\displaystyle P(x,y)=x^{2}-4xy^{4,5}\,}"></span></li> <li><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^{\frac {3}{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"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{\frac {3}{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b713fdfbd82cfb06516a48d39a710a3dd365fed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.435ex; height:4.009ex;" alt="{\displaystyle P(x)=x^{\frac {3}{2}}\,}"></span></li> <li><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)={\sqrt {x}}=x^{\frac {1}{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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>x</mi> </msqrt> </mrow> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)={\sqrt {x}}=x^{\frac {1}{2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c134587b95b5b1a5aae4a3d8493dc444fb321010" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.799ex; height:4.176ex;" alt="{\displaystyle P(x)={\sqrt {x}}=x^{\frac {1}{2}}\,}"></span></li> <li><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)=(5+y)^{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> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mo>+</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=(5+y)^{x}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a467017e20a40cdc89e3afc255c6dcb1a7be03fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.51ex; height:2.843ex;" alt="{\displaystyle P(x)=(5+y)^{x}\,}"></span></li> <li><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)={\frac {1}{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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)={\frac {1}{x+2}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3787ec059e2cc8f4fa16b72993fde0893581b22b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.539ex; height:5.343ex;" alt="{\displaystyle P(x)={\frac {1}{x+2}}\,}"></span></li></ul> <p>Gaur egun polinomioak adierazteko erabiltzen dugun notazioa XV. mendean garatu zen. Notazio hori baino lehen, hitzen bitartez idazten ziren. <i>"<a href="https://en.wikipedia.org/wiki/The_Nine_Chapters_on_the_Mathematical_Art" class="extiw" title="en:The Nine Chapters on the Mathematical Art">Arithmetic in Nine Sections</a>"</i> aljebra-liburuan, adibidez, ikus dezakegu hitzezko notazio hori. <i>La géometrie</i> liburuan (1637), <a href="/wiki/Ren%C3%A9_Descartes" title="René Descartes">René Descartes</a> matematikariak proposatu zuen: konstanteak alfabetoaren lehenengo hizkiez adieraztea (a, b, c, d...) eta ezezagunak azken hizkiez (x,y,z). </p><p>Batugaiak lau baino gutxiago badira, izen hauek jasotzen dituzte polinomioek: <a href="/wiki/Monomio" title="Monomio">monomio</a> (batugai bakarra), binomio (bi batugai) eta trinomio (hiru batugai). </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historia">Historia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=1" title="Aldatu atal hau: «Historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=1" title="Edit section's source code: Historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ekuazio aljebraikoak ebaztea edo polinomioen erroak zehaztea da matematikako arazo zaharrenetakoa. Hala ere, gaur egun erabiltzen dugun idazkera dotore eta praktikoa XV. Mendetik aurrera garatu zen. </p><p>Moskuko papiroaren 14. probleman (K.a. 1890. urtea) piramide enbor laukilarraren bolumena kalkulatzea eskatzen da. Eskribauak pausoak azaltzen ditu: 2 eta 4 karratuak, 2 4 bider, aurreko emaitzak gehitu eta 6 (h) -ren herena biderkatu; Honela esanez amaitzen du: "Ikusi, 56 dira, ondo kalkulatu duzu". Egungo notazio aljebraikoan honakoa litzateke: V = h (t² + b² + tb) / 3, lau aldagaiko polinomioa (V, h, t, b), hiru jakinda, laugarren aldagaia lortzeko aukera ematen duena. </p><p>Polinomio batzuek, hala nola P (x) = x² + 1, ez dute zenbaki erreala den erroa. Hala ere, erro posibleen multzoa zenbaki konplexuetara hedatzen bada, polinomio (ez konstante) orok du erro bat: hori da <a href="/wiki/Aljebra_lineal" title="Aljebra lineal">aljebra</a>ko oinarrizko <a href="/wiki/Teorema" title="Teorema">teorema</a>ren adierazpena. </p><p>Desberdintasunak daude erroen hurbilketaren eta haientzako formula konkretuen aurkikuntzaren artean. Mendetik laugarren gradurainoko polinomioen formak ezagutzen dira (ikus <a href="/wiki/Bigarren_mailako_ekuazio" title="Bigarren mailako ekuazio">ekuazio koadratiko</a>a, <a href="/wiki/Girolamo_Cardano" title="Girolamo Cardano">Gerolamo Cardano</a>, <a href="/wiki/Niccol%C3%B2_Fontana_Tartaglia" title="Niccolò Fontana Tartaglia">Niccolò Fontana Tartaglia</a>). Baina, bosgarren mailako polinomioen formulak konponezinak izan ziren ikertzaileentzat denbora luzez. <a href="/wiki/1824" title="1824">1824</a>an, <a href="/wiki/Niels_Henrik_Abel" title="Niels Henrik Abel">Niels Henrik Abel</a>ek erakutsi zuen ezin dela bosgarren graduko edo gehiagoko polinomioetarako formula orokorrik egon (ikus <a href="/w/index.php?title=Abel-Ruffini_teorema&action=edit&redlink=1" class="new" title="Abel-Ruffini teorema (sortu gabe)">Abel-Ruffini teorema</a>). Emaitza horrek Polinomioen erroen arteko erlazioen azterketa zehatzaz arduratzen den <a href="/wiki/%C3%89variste_Galois" title="Évariste Galois">Galois</a> teoriaren hasiera izan zen. </p><p><a href="/wiki/Charles_Babbage" title="Charles Babbage">Charles Babbage</a>ren motor diferentziala funtzio logaritmikoen eta diferentzialen balioen taulak automatikoki sortzeko diseinatu zen, puntu askotan hurbilketa polinomikoak ebaluatuz, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Isaac Newton</a>en desberdintasunen metodoa erabiliz. </p> <div class="mw-heading mw-heading2"><h2 id="Monomioa">Monomioa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=2" title="Aldatu atal hau: «Monomioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=2" title="Edit section's source code: Monomioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_2" poster="//upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Zer_dira_monomioak%3F.webm/220px--Zer_dira_monomioak%3F.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="124" data-durationhint="236" data-mwtitle="Zer_dira_monomioak?.webm" data-mwprovider="wikimediacommons" resource="/wiki/Fitxategi:Zer_dira_monomioak%3F.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/9/9a/Zer_dira_monomioak%3F.webm" type="video/webm; codecs="vp9, opus"" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/9/9a/Zer_dira_monomioak%3F.webm/Zer_dira_monomioak%3F.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption><b>Monomioak</b> ulertzeko bideoa.<hr /><span style="color: #777;"><figure class="mw-halign-left" typeof="mw:File"><a href="https://www.jakindun.com/" rel="nofollow"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/33px-Jakindun_logoa.png" decoding="async" width="33" height="33" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/50px-Jakindun_logoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/66px-Jakindun_logoa.png 2x" data-file-width="225" data-file-height="225" /></a><figcaption></figcaption></figure>Bideo hau Jakindun elkarteak egin du. Gehiago dituzu eskuragarri <a rel="nofollow" class="external text" href="https://www.jakindun.com/">euren gunean</a>. Bideoak dituzten artikulu guztiak ikus ditzakezu <a href="/wiki/Kategoria:Jakindunen_bideoak_dituzten_artikuluak" title="Kategoria:Jakindunen bideoak dituzten artikuluak"><b>hemen</b></a>.</span></figcaption></figure><p><b></b> </p><p><a href="/wiki/Monomio" title="Monomio">Monomioa</a><b> gai bakarreko <a href="/wiki/Adierazpen_(matematika)" class="mw-redirect" title="Adierazpen (matematika)">adierazpen aljebraikoa</a> da. Gai hori <a href="/wiki/Zenbaki" title="Zenbaki">zenbakien</a> eta <a href="/wiki/Aldagai_(matematika)" title="Aldagai (matematika)">aldagaien</a> arteko <a href="/wiki/Biderketa" title="Biderketa">biderkadura</a> da. Zenbakiei <a href="/wiki/Koefiziente_(matematika)" title="Koefiziente (matematika)">koefiziente</a> ere baderitze eta monomioaren hasieran idazten ohi dira. </b> </p><p>Notazioa: <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_{k}x^{k}\mid \forall _{k}\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>∣</mo> <msub> <mi mathvariant="normal">∀</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k}x^{k}\mid \forall _{k}\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcadc9bb65342e0921cb3be122792d9cc49422ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.574ex; height:3.176ex;" alt="{\displaystyle a_{k}x^{k}\mid \forall _{k}\in \mathbb {N} }"></span> </p><p>Era berean, monomio baten maila aldagaiaren berretzailea da, konstante ez nulua bada, maila 0 izango delarik. </p><p>Adibidez: </p> <ul><li><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\cdot b=ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b=ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41854002ecf5428f51becf70778bcb176048324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.232ex; height:2.176ex;" alt="{\displaystyle a\cdot b=ab}"></span></li> <li><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\cdot x\cdot y=x^{2}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <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 x\cdot x\cdot y=x^{2}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1081c7cbd49ebf88cf4db8a5433192cc6cf55f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.811ex; height:3.009ex;" alt="{\displaystyle x\cdot x\cdot y=x^{2}y}"></span></li> <li><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\cdot q\cdot q\cdot q\cdot r=pq^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>r</mi> <mo>=</mo> <mi>p</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\cdot q\cdot q\cdot q\cdot r=pq^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/133bf8877bf3aeb7fe65a20776d592cb768e47cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:19.683ex; height:3.009ex;" alt="{\displaystyle p\cdot q\cdot q\cdot q\cdot r=pq^{3}r}"></span></li> <li><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 2\cdot a\cdot b=2ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>⋅</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\cdot a\cdot b=2ab}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c81b459c23f9be7c5a7bb0eb760215cfcc6dca9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.236ex; height:2.176ex;" alt="{\displaystyle 2\cdot a\cdot b=2ab}"></span></li> <li><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 3,15\cdot x\cdot x\cdot y=3,15x^{2}y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>,</mo> <mn>15</mn> <mo>⋅</mo> <mi>x</mi> <mo>⋅</mo> <mi>x</mi> <mo>⋅</mo> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>15</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 3,15\cdot x\cdot x\cdot y=3,15x^{2}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30834287b8374f852dfd6e8ba39d7eb787599b26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.533ex; height:3.009ex;" alt="{\displaystyle 3,15\cdot x\cdot x\cdot y=3,15x^{2}y}"></span></li> <li><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 {\sqrt {5}}p\cdot q\cdot q\cdot q\cdot r={\sqrt {5}}pq^{3}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mi>p</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>q</mi> <mo>⋅</mo> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> <mi>p</mi> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {5}}p\cdot q\cdot q\cdot q\cdot r={\sqrt {5}}pq^{3}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9b4b995e201cd845a8cabb75479fbdb6acabf8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.79ex; height:3.009ex;" alt="{\displaystyle {\sqrt {5}}p\cdot q\cdot q\cdot q\cdot r={\sqrt {5}}pq^{3}r}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Aplikazioak">Aplikazioak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=3" title="Aldatu atal hau: «Aplikazioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=3" title="Edit section's source code: Aplikazioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomioa funtsezko kontzeptua da <a href="/wiki/Aljebra" title="Aljebra">aljebran</a>. <a href="/wiki/Matematika" title="Matematika">Matematikan</a>, <a href="/wiki/Funtzio_(argipena)" class="mw-disambig" title="Funtzio (argipena)">funtzioak</a> hurbiltzeko erabiltzen dira. <a href="/wiki/Kimika" title="Kimika">Kimikan</a> eta <a href="/wiki/Fisika" title="Fisika">fisikan</a> aplikazio handiak dituzte, baita <a href="/wiki/Ekonomia" title="Ekonomia">ekonomian</a> eta <a href="/wiki/Kriptografia" title="Kriptografia">kriptografian</a> ere. </p> <div class="mw-heading mw-heading2"><h2 id="Definizio_aljebraikoa">Definizio aljebraikoa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=4" title="Aldatu atal hau: «Definizio aljebraikoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=4" title="Edit section's source code: Definizio aljebraikoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Aldagai_bakarreko_polinomioak">Aldagai bakarreko polinomioak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=5" title="Aldatu atal hau: «Aldagai bakarreko polinomioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=5" title="Edit section's source code: Aldagai bakarreko polinomioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomioari, ezezagun bakarra daukanean, aldagai bakarreko polinomio deritzo. Adibidez, </p> <ul><li><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+1}"> <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>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f85c17afa9136416dccb6a572e411d29bec604eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.315ex; height:2.843ex;" alt="{\displaystyle P(x)=x+1}"></span></li></ul> <ul><li><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)=2x_{}^{2}+4x+1}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{2}+4x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/898e80ec0eebddab89f193b8402cc92a737cdec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.865ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{2}+4x+1}"></span></li> <li><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_{}^{5}-5x_{}^{4}+2x_{}^{3}-x_{}^{2}+7x+1}"> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>7</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x_{}^{5}-5x_{}^{4}+2x_{}^{3}-x_{}^{2}+7x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/209e9997d8d59ff60df2c7787832662af35083fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.7ex; height:3.176ex;" alt="{\displaystyle P(x)=x_{}^{5}-5x_{}^{4}+2x_{}^{3}-x_{}^{2}+7x+1}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Aldagai_anitzeko_polinomioak">Aldagai anitzeko polinomioak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=6" title="Aldatu atal hau: «Aldagai anitzeko polinomioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=6" title="Edit section's source code: Aldagai anitzeko polinomioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ezezagun bat baino gehiago daukanean, polinomioari aldagai anitzeko polinomio deritzo. Adibidez, </p> <ul><li><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+y}"> <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>x</mi> <mo>+</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x+y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61b53bff9f2d8a30fa3d6ce6b0dd70a3e7d02a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.308ex; height:2.843ex;" alt="{\displaystyle P(x)=x+y}"></span></li></ul> <ul><li><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)=2x_{}^{2}y-3xy+1}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=2x_{}^{2}y-3xy+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34b2e8178c7a3ca2fac6498e97ccb25833099834" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.365ex; height:3.176ex;" alt="{\displaystyle P(x,y)=2x_{}^{2}y-3xy+1}"></span></li> <li><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(y,z)=y_{}^{3}-yz_{}^{2}+3z_{}^{3}-y{}^{2}z+2yz+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mi>y</mi> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mi>y</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>+</mo> <mn>2</mn> <mi>y</mi> <mi>z</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(y,z)=y_{}^{3}-yz_{}^{2}+3z_{}^{3}-y{}^{2}z+2yz+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc3eec671c5609823d20dafcec0aea50ebd1804e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.821ex; height:3.176ex;" alt="{\displaystyle P(y,z)=y_{}^{3}-yz_{}^{2}+3z_{}^{3}-y{}^{2}z+2yz+1}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Polinomioaren_maila">Polinomioaren maila</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=7" title="Aldatu atal hau: «Polinomioaren maila»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=7" title="Edit section's source code: Polinomioaren maila"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Notazioa">Notazioa</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=8" title="Aldatu atal hau: «Notazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=8" title="Edit section's source code: Notazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomio baten maila adierazteko deg laburpena erabiltzen da.Beraz, <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 degf}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>e</mi> <mi>g</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle degf}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b0d1911b34044b381f01b49a72676ffdaf8ccfe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.694ex; height:2.509ex;" alt="{\displaystyle degf}"></span> da <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> polinomioaren maila. </p> <div class="mw-heading mw-heading4"><h4 id="Monomioak">Monomioak</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=9" title="Aldatu atal hau: «Monomioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=9" title="Edit section's source code: Monomioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Ezezagun bakarreko monomioetan, aldagaiaren berretzailea da monomioaren maila. </p><p>Ezezagun bat baino gehiagoko monomioetan, aldiz, aldagai guztien berretzaileen batura da maila. Adibidez, </p> <ul><li><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)=3x}"> <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> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1dbd1ddf008e043915e7bac9c83ece0222b78fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle P(x)=3x}"></span><i>,</i> monomioaren maila 1 da.</li> <li><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)=2x_{}^{5}}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b60a24b81ec63388e387ef2bac12ce2a069c6bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.529ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{5}}"></span>, monomioaren maila 5 da.</li> <li><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)=3xz_{}^{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> <mn>3</mn> <mi>x</mi> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=3xz_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7752a216b934309ef8c7e63034c5782d32aad35a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.62ex; height:3.176ex;" alt="{\displaystyle P(x)=3xz_{}^{2}}"></span>, polinomioaren maila 3 da.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Polinomioak">Polinomioak</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=10" title="Aldatu atal hau: «Polinomioak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=10" title="Edit section's source code: Polinomioak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomio baten maila bere monomio artean maila altuena duenarena maila da. <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_{0}+a_{1}x+...+a_{n}x^{n}|a_{1}\neq 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"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</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> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=a_{0}+a_{1}x+...+a_{n}x^{n}|a_{1}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e2ecae990ebf85cab20de4f972a0cab369ee6c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:35.141ex; height:2.843ex;" alt="{\displaystyle p(x)=a_{0}+a_{1}x+...+a_{n}x^{n}|a_{1}\neq 0}"></span> </p><p>Orduan, <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 degp=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>e</mi> <mi>g</mi> <mi>p</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle degp=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdaa099d4d7a596f9725620acf7d98f2cc135d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.078ex; height:2.509ex;" alt="{\displaystyle degp=n}"></span> </p> <ul><li><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)=3x_{}^{2}+2x+1}"> <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> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=3x_{}^{2}+2x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9972c235fb32dab35a2101855155e278a0f9468a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.865ex; height:3.176ex;" alt="{\displaystyle P(x)=3x_{}^{2}+2x+1}"></span><i>,</i> polinomioaren maila 2 da.</li> <li><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_{}^{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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x_{}^{3}-3x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92ea6ed7bd1eecd4d4168a3020855985a638a359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.702ex; height:3.176ex;" alt="{\displaystyle P(x)=x_{}^{3}-3x+2}"></span>, polinomioaren maila 3 da.</li> <li><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)=4x_{}^{4}+4x+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> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=4x_{}^{4}+4x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f69955d40879331d5ec3ef9e05cadca0382ad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.865ex; height:3.176ex;" alt="{\displaystyle P(x)=4x_{}^{4}+4x+2}"></span>, polinomioaren maila 4 da.</li> <li><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_{}^{7}+2x_{}^{5}-4x_{}^{3}-6x+1}"> <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> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x_{}^{7}+2x_{}^{5}-4x_{}^{3}-6x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53c5144f65ff1e05051c2893d27bc46433d0bded" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.476ex; height:3.176ex;" alt="{\displaystyle P(x)=x_{}^{7}+2x_{}^{5}-4x_{}^{3}-6x+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 P(x)=3x}"> <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> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1dbd1ddf008e043915e7bac9c83ece0222b78fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle P(x)=3x}"></span>, polinomioaren maila 7 da.</li></ul> <p>Bereiz ditzakegu bi kasu berezi: </p> <ul><li>Polinomio konstantea: <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)=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> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99803f5da907baa1f9835df5f8509080ff2bf3cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle P(x)=2}"></span>, monomioaren maila 0 da (<i>x</i><sup>0</sup>=1 baita); beraz, zero mailako polinomioak konstante ez-nuluak izango dira.</li> <li>Polinomio nulua: <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)=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> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b206fb7efdcf09cecb110d09e4543295673ef4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.145ex; height:2.843ex;" alt="{\displaystyle P(x)=0}"></span>, polinomioaren maila <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 \scriptstyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ce3c1e43594068ee5147c2e49e8feebcefacb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.922ex; height:1.509ex;" alt="{\displaystyle \scriptstyle -\infty }"></span> da.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Eragiketak_polinomioekin">Eragiketak polinomioekin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=11" title="Aldatu atal hau: «Eragiketak polinomioekin»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=11" title="Edit section's source code: Eragiketak polinomioekin"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Polinomioa_puntu_batean_ebaluatzea">Polinomioa puntu batean ebaluatzea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=12" title="Aldatu atal hau: «Polinomioa puntu batean ebaluatzea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=12" title="Edit section's source code: Polinomioa puntu batean ebaluatzea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edozein polinomio p puntu batean ebaluatzeko, indeterminatuaren lekuan p puntua ordezkatzea besterik ez da egin behar. Lortzen dugun emaitzari polinomioaren zenbakizko balioa<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> deritzo. Adibidez, </p> <ul><li><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)=4x_{}^{4}+4x+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> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=4x_{}^{4}+4x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61f69955d40879331d5ec3ef9e05cadca0382ad9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.865ex; height:3.176ex;" alt="{\displaystyle P(x)=4x_{}^{4}+4x+2}"></span> polinomioa x=2 puntuan ebaluatzeko, zera egingo dugu: <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(2)=4\cdot (2)_{}^{4}+4\cdot (2)+2=74}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mn>74</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(2)=4\cdot (2)_{}^{4}+4\cdot (2)+2=74}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16c220b2e8698929d3914e79f42d317540c8677d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.763ex; height:3.176ex;" alt="{\displaystyle P(2)=4\cdot (2)_{}^{4}+4\cdot (2)+2=74}"></span>; kasu honetan, x=2 puntuari dagokion P(x) polinomioaren zenbakizko balioa 74 da.</li> <li><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)=5x_{}^{2}y+3xy_{}^{2}-2xy+1}"> <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> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=5x_{}^{2}y+3xy_{}^{2}-2xy+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad23b24a03409ffce112f226396cb3e81339a5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.912ex; height:3.176ex;" alt="{\displaystyle P(x,y)=5x_{}^{2}y+3xy_{}^{2}-2xy+1}"></span> polinomioa (x,y) = (-2,1) puntuan ebaluatzeko: <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(-2,1)=5\cdot (-2)_{}^{2}\cdot (1)+3\cdot (-2)\cdot (1)_{}^{2}-2\cdot (-2)\cdot (1)+1=19}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mn>19</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(-2,1)=5\cdot (-2)_{}^{2}\cdot (1)+3\cdot (-2)\cdot (1)_{}^{2}-2\cdot (-2)\cdot (1)+1=19}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1039368d81732163a1eac1d1009978c9ac8bee7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:65.853ex; height:3.176ex;" alt="{\displaystyle P(-2,1)=5\cdot (-2)_{}^{2}\cdot (1)+3\cdot (-2)\cdot (1)_{}^{2}-2\cdot (-2)\cdot (1)+1=19}"></span> ; kasu honetan, (x,y)=(-2,1) puntuari dagokion P(x,y) polinomioaren zenbakizko balioa 19 da.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Batuketa_eta_kenketa">Batuketa eta kenketa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=13" title="Aldatu atal hau: «Batuketa eta kenketa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=13" title="Edit section's source code: Batuketa eta kenketa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Monomioen <a href="/wiki/Batuketa" title="Batuketa">batuketa</a> edo <a href="/wiki/Kenketa" title="Kenketa">kenketa</a> egin ahal izateko, antzekoak izan behar dira monomioak; hau da, gai aljebraiko (<a href="/wiki/Aldagai_(matematika)" title="Aldagai (matematika)">aldagaien</a> zatia) berbera izan behar dute. Kasu horretan, gai aljebraikoa mantentzen da eta koefizienteen batuketa edo kenketa egiten da. </p><p>Batuketa:<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 f(x)+g(x)=(a_{0}+b_{0})+(a_{1}+b_{1})+...+(a_{n}+b_{n})x^{n}+...+b_{m}x^{m},n\leq m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> <mi>n</mi> <mo>≤</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)+g(x)=(a_{0}+b_{0})+(a_{1}+b_{1})+...+(a_{n}+b_{n})x^{n}+...+b_{m}x^{m},n\leq m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f5c64b671e2ea4c54a87a1361ff952ec8e3b172" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:73.966ex; height:2.843ex;" alt="{\displaystyle f(x)+g(x)=(a_{0}+b_{0})+(a_{1}+b_{1})+...+(a_{n}+b_{n})x^{n}+...+b_{m}x^{m},n\leq m}"></span> </p><p>Adibidez: </p> <dl><dd><ul><li><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)=2x}"> <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> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aaec25ae9d28ccdc3f26f790de53eeb2f3029186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.475ex; height:2.843ex;" alt="{\displaystyle P(x)=2x}"></span> eta <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)=3x}"> <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> <mn>3</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=3x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee4bf3c887c20424247d84e4ec680f2f8a05659d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.568ex; height:2.843ex;" alt="{\displaystyle Q(x)=3x}"></span> monomioak izanda, <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)=2x+3x=5x}"> <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> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mn>5</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)+Q(x)=2x+3x=5x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4c83f814f8299fffa0b5ba320caa2dd4bd7e6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.216ex; height:2.843ex;" alt="{\displaystyle P(x)+Q(x)=2x+3x=5x}"></span> eta <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)=2x-3x=-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> <mo>−</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)-Q(x)=2x-3x=-x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71b30dc4f381b9afeddccdc3b3a59cb6ee173577" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.862ex; height:2.843ex;" alt="{\displaystyle P(x)-Q(x)=2x-3x=-x}"></span></li> <li><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)=2x_{}^{5}}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b60a24b81ec63388e387ef2bac12ce2a069c6bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.529ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{5}}"></span> eta <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)=5x_{}^{5}}"> <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> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=5x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864d30ca19ffdddd0026c15aaaa068262cb69200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.622ex; height:3.176ex;" alt="{\displaystyle Q(x)=5x_{}^{5}}"></span>izanda, <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)=2x_{}^{5}+5x_{}^{5}=7x_{}^{5}}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>+</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mn>7</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)+Q(x)=2x_{}^{5}+5x_{}^{5}=7x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df79d0475a83ec268de216eed9dfea44bd764835" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.379ex; height:3.176ex;" alt="{\displaystyle P(x)+Q(x)=2x_{}^{5}+5x_{}^{5}=7x_{}^{5}}"></span>eta <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)=2x_{}^{5}-5x_{}^{5}=-3x_{}^{5}}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)-Q(x)=2x_{}^{5}-5x_{}^{5}=-3x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a0daf45b43a43ee18a7efb8edaa511bf2e1452" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.187ex; height:3.176ex;" alt="{\displaystyle P(x)-Q(x)=2x_{}^{5}-5x_{}^{5}=-3x_{}^{5}}"></span></li> <li><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)+Q(x,y)=3x_{}^{2}y+4x_{}^{2}y=7x_{}^{2}y}"> <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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>+</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>=</mo> <mn>7</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)+Q(x,y)=3x_{}^{2}y+4x_{}^{2}y=7x_{}^{2}y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56feb918c6f03db95082c065fdd2a2a6cd704ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.224ex; height:3.176ex;" alt="{\displaystyle P(x,y)+Q(x,y)=3x_{}^{2}y+4x_{}^{2}y=7x_{}^{2}y}"></span></li> <li><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}y-2xy_{}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{}^{2}y-2xy_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c661e07ff71206cbe07dbd890328fc2de03e7d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.087ex; height:3.009ex;" alt="{\displaystyle x_{}^{2}y-2xy_{}^{2}}"></span>(ezin dira batu monomioak, antzekoak ez direlako)</li></ul></dd> <dd></dd></dl> <p>Polinomioen arteko <a href="/wiki/Batuketa" title="Batuketa">batuketa</a> edo <a href="/wiki/Kenketa" title="Kenketa">kenketa</a> egiteko, antzekoak diren monomioak batu edo kendu behar ditugu. Adibidez, </p> <ul><li><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)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{2}+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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{2}+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8c542adc300ecf23c37a716c38cda3a99d448e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.304ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{2}+3}"></span> eta <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)=2x_{}^{6}-3x_{}^{5}+4x_{}^{4}+x_{}^{3}+2x_{}^{2}-3x+2}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=2x_{}^{6}-3x_{}^{5}+4x_{}^{4}+x_{}^{3}+2x_{}^{2}-3x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/923d2a976385ab9faf7c0ca8eaf4e28c934413d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.342ex; height:3.176ex;" alt="{\displaystyle Q(x)=2x_{}^{6}-3x_{}^{5}+4x_{}^{4}+x_{}^{3}+2x_{}^{2}-3x+2}"></span> izanda,</li></ul> <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 P(x)+Q(x)=2x_{}^{7}+(3x_{}^{6}+2x_{}^{6})+(x_{}^{5}-3x_{}^{5})+(-6x_{}^{4}+4x_{}^{4})+(2x_{}^{3}+x_{}^{3})+(-5x_{}^{2}+2x_{}^{2})-3x+(3+2)=2x_{}^{7}+5x_{}^{6}-2x_{}^{5}-2x_{}^{4}+3x_{}^{3}-3x_{}^{2}-3x+5}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)+Q(x)=2x_{}^{7}+(3x_{}^{6}+2x_{}^{6})+(x_{}^{5}-3x_{}^{5})+(-6x_{}^{4}+4x_{}^{4})+(2x_{}^{3}+x_{}^{3})+(-5x_{}^{2}+2x_{}^{2})-3x+(3+2)=2x_{}^{7}+5x_{}^{6}-2x_{}^{5}-2x_{}^{4}+3x_{}^{3}-3x_{}^{2}-3x+5}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5358179ad3b329e6176ab54cfc6b7ac71787491e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:156.614ex; height:3.176ex;" alt="{\displaystyle P(x)+Q(x)=2x_{}^{7}+(3x_{}^{6}+2x_{}^{6})+(x_{}^{5}-3x_{}^{5})+(-6x_{}^{4}+4x_{}^{4})+(2x_{}^{3}+x_{}^{3})+(-5x_{}^{2}+2x_{}^{2})-3x+(3+2)=2x_{}^{7}+5x_{}^{6}-2x_{}^{5}-2x_{}^{4}+3x_{}^{3}-3x_{}^{2}-3x+5}"></span> eta </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 P(x)-Q(x)=2x_{}^{7}+(3x_{}^{6}-2x_{}^{6})+(x_{}^{5}+3x_{}^{5})+(-6x_{}^{4}-4x_{}^{4})+(2x_{}^{3}-x_{}^{3})+(-5x_{}^{2}-2x_{}^{2})+3x+(3-2)=2x_{}^{7}+x_{}^{6}+4x_{}^{5}-10x_{}^{4}+x_{}^{3}-7x_{}^{2}+3x+1}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>6</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>−</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>10</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>7</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)-Q(x)=2x_{}^{7}+(3x_{}^{6}-2x_{}^{6})+(x_{}^{5}+3x_{}^{5})+(-6x_{}^{4}-4x_{}^{4})+(2x_{}^{3}-x_{}^{3})+(-5x_{}^{2}-2x_{}^{2})+3x+(3-2)=2x_{}^{7}+x_{}^{6}+4x_{}^{5}-10x_{}^{4}+x_{}^{3}-7x_{}^{2}+3x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c81f091e6c79db6e96eaa779621dbfe6bae40e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:155.452ex; height:3.176ex;" alt="{\displaystyle P(x)-Q(x)=2x_{}^{7}+(3x_{}^{6}-2x_{}^{6})+(x_{}^{5}+3x_{}^{5})+(-6x_{}^{4}-4x_{}^{4})+(2x_{}^{3}-x_{}^{3})+(-5x_{}^{2}-2x_{}^{2})+3x+(3-2)=2x_{}^{7}+x_{}^{6}+4x_{}^{5}-10x_{}^{4}+x_{}^{3}-7x_{}^{2}+3x+1}"></span> </p> <ul><li><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)=2x_{}^{2}y-3xy-xy_{}^{2}+1}"> <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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=2x_{}^{2}y-3xy-xy_{}^{2}+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d7a147a88db374ca16b2377cdd77d278d82c84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.75ex; height:3.176ex;" alt="{\displaystyle P(x,y)=2x_{}^{2}y-3xy-xy_{}^{2}+1}"></span> eta <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,y)=x_{}^{2}y+2xy+3xy_{}^{2}-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x,y)=x_{}^{2}y+2xy+3xy_{}^{2}-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/270847b402e359f50ead7e3f29a435dd253ff8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.843ex; height:3.176ex;" alt="{\displaystyle Q(x,y)=x_{}^{2}y+2xy+3xy_{}^{2}-2}"></span> izanda,</li></ul> <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 P(x,y)+Q(x,y)=(2x_{}^{2}y+x_{}^{2}y)+(-3xy+2xy)+(-xy_{}^{2}+3xy_{}^{2})+(1-2)=3x_{}^{2}y-xy+2xy_{}^{2}-1}"> <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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>2</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)+Q(x,y)=(2x_{}^{2}y+x_{}^{2}y)+(-3xy+2xy)+(-xy_{}^{2}+3xy_{}^{2})+(1-2)=3x_{}^{2}y-xy+2xy_{}^{2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8190bb2e94c49ea053cc0dd2165d6d42518548c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.704ex; height:3.176ex;" alt="{\displaystyle P(x,y)+Q(x,y)=(2x_{}^{2}y+x_{}^{2}y)+(-3xy+2xy)+(-xy_{}^{2}+3xy_{}^{2})+(1-2)=3x_{}^{2}y-xy+2xy_{}^{2}-1}"></span> eta </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 P(x,y)-Q(x,y)=(2x_{}^{2}y-x_{}^{2}y)+(-3xy-2xy)+(-xy_{}^{2}-3xy_{}^{2})+(1+2)=x_{}^{2}y-5xy-4xy_{}^{2}+3}"> <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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mn>5</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>4</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)-Q(x,y)=(2x_{}^{2}y-x_{}^{2}y)+(-3xy-2xy)+(-xy_{}^{2}-3xy_{}^{2})+(1+2)=x_{}^{2}y-5xy-4xy_{}^{2}+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22e52300989ad26547f5b31186e6ac386f30a164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:101.704ex; height:3.176ex;" alt="{\displaystyle P(x,y)-Q(x,y)=(2x_{}^{2}y-x_{}^{2}y)+(-3xy-2xy)+(-xy_{}^{2}-3xy_{}^{2})+(1+2)=x_{}^{2}y-5xy-4xy_{}^{2}+3}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Biderketa">Biderketa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=14" title="Aldatu atal hau: «Biderketa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=14" title="Edit section's source code: Biderketa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Monomioen arteko biderketa egiteko, koefizienteak biderkatu eta indeterminatu berdinen mailak batu behar ditugu. </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 f(x)=\sum _{i=0}a_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> </munder> <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 f(x)=\sum _{i=0}a_{i}x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/257f8207578a8135c7c5efee86d66cb57ecf62d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.417ex; height:5.509ex;" alt="{\displaystyle f(x)=\sum _{i=0}a_{i}x^{i}}"></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 g(x)=\sum _{i=0}b_{i}x^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>b</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 g(x)=\sum _{i=0}b_{i}x^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f044cc9251d79e272f94c4075f5a77a9265d93a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.022ex; height:5.509ex;" alt="{\displaystyle g(x)=\sum _{i=0}b_{i}x^{i}}"></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 f(x)g(x)=\sum _{k\geq 0}(\sum _{i+j=k}a_{i}b_{i})x^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mo stretchy="false">(</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>=</mo> <mi>k</mi> </mrow> </munder> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)g(x)=\sum _{k\geq 0}(\sum _{i+j=k}a_{i}b_{i})x^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6631a01e01e4358dc60f4bad3ef0f0296d614168" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.226ex; height:5.843ex;" alt="{\displaystyle f(x)g(x)=\sum _{k\geq 0}(\sum _{i+j=k}a_{i}b_{i})x^{k}}"></span> </p><p>Adibidez, </p> <ul><li><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)=4x_{}^{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> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=4x_{}^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4296b38556270995e01db11ec2dcb5d5204dbb2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.529ex; height:3.176ex;" alt="{\displaystyle P(x)=4x_{}^{3}}"></span>eta <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)=3x_{}^{5}}"> <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> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=3x_{}^{5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0845b943aa1189158ca02903f33b2d941ef46ae8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.622ex; height:3.176ex;" alt="{\displaystyle Q(x)=3x_{}^{5}}"></span> monomioak izanda, <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)_{}^{}=4x_{}^{3}\cdot 3x_{}^{5}=4\cdot 3\cdot x_{}^{3+5}=12x_{}^{8}}"> <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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>⋅</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mn>5</mn> </mrow> </msubsup> <mo>=</mo> <mn>12</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)Q(x)_{}^{}=4x_{}^{3}\cdot 3x_{}^{5}=4\cdot 3\cdot x_{}^{3+5}=12x_{}^{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f1987d6d6daee497e0f9377bb3df134889e2511" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.805ex; height:3.176ex;" alt="{\displaystyle P(x)Q(x)_{}^{}=4x_{}^{3}\cdot 3x_{}^{5}=4\cdot 3\cdot x_{}^{3+5}=12x_{}^{8}}"></span></li> <li><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)=4x_{}^{3}y_{}^{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> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=4x_{}^{3}y_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c257e4fdeaf556cd1269db5cc6cb5cf7ba2569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.744ex; height:3.176ex;" alt="{\displaystyle P(x)=4x_{}^{3}y_{}^{2}}"></span>eta <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)=3x_{}^{5}y_{}^{8}}"> <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> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=3x_{}^{5}y_{}^{8}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24398f51b5599377512a736166b0c3918d3917f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.837ex; height:3.176ex;" alt="{\displaystyle Q(x)=3x_{}^{5}y_{}^{8}}"></span> monomioak izanda, <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)_{}^{}=4x_{}^{3}y_{}^{2}\cdot 3x_{}^{5}y_{}^{8}=4\cdot 3\cdot x_{}^{3+5}\cdot y_{}^{2+8}=12x_{}^{8}y_{}^{10}}"> <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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>⋅</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msubsup> <mo>=</mo> <mn>4</mn> <mo>⋅</mo> <mn>3</mn> <mo>⋅</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>+</mo> <mn>5</mn> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>+</mo> <mn>8</mn> </mrow> </msubsup> <mo>=</mo> <mn>12</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)Q(x)_{}^{}=4x_{}^{3}y_{}^{2}\cdot 3x_{}^{5}y_{}^{8}=4\cdot 3\cdot x_{}^{3+5}\cdot y_{}^{2+8}=12x_{}^{8}y_{}^{10}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d436da419a924f6be30f28e13ebd130f20f195d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.266ex; height:3.176ex;" alt="{\displaystyle P(x)Q(x)_{}^{}=4x_{}^{3}y_{}^{2}\cdot 3x_{}^{5}y_{}^{8}=4\cdot 3\cdot x_{}^{3+5}\cdot y_{}^{2+8}=12x_{}^{8}y_{}^{10}}"></span></li></ul> <p>Bi polinomioen arteko biderketa egiteko, polinomio baten gai bakoitza beste polinomioaren gai guztiekin biderkatu behar da, eta ondoren, maila bereko terminoak batu edo kendu. Adibidez, </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 P(x)=(2x_{}^{3}+4x+1)}"> <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> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=(2x_{}^{3}+4x+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d429f9f35b54f03fda34a5512b7bfa373bf2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.674ex; height:3.176ex;" alt="{\displaystyle P(x)=(2x_{}^{3}+4x+1)}"></span>eta <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)_{}^{}=(5x^{2}+3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)_{}^{}=(5x^{2}+3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fecc64134754c09683a0520c1bf4f729e9efb1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.434ex; height:3.176ex;" alt="{\displaystyle Q(x)_{}^{}=(5x^{2}+3)}"></span> polinomioak izanda,<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)_{}^{}=(2x_{}^{3}+4x+1)(5x^{2}+3)=(2x_{}^{3}+4x+1)(5x^{2})+(2x^{3}+4x+1)(3)=(2x_{}^{3}\cdot 5x^{2}+4x\cdot 5x^{2}+1\cdot 5x^{2})+(2x^{3}\cdot 3+4x\cdot 3+1\cdot 3)=10x_{}^{5}+20x^{3}+5x^{2}+6x^{3}+12x+3=10x_{}^{5}+26x^{3}+5x^{2}+12x+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> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msubsup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> </mrow> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>⋅</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>⋅</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>10</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>+</mo> <mn>20</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mo>=</mo> <mn>10</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>+</mo> <mn>26</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)Q(x)_{}^{}=(2x_{}^{3}+4x+1)(5x^{2}+3)=(2x_{}^{3}+4x+1)(5x^{2})+(2x^{3}+4x+1)(3)=(2x_{}^{3}\cdot 5x^{2}+4x\cdot 5x^{2}+1\cdot 5x^{2})+(2x^{3}\cdot 3+4x\cdot 3+1\cdot 3)=10x_{}^{5}+20x^{3}+5x^{2}+6x^{3}+12x+3=10x_{}^{5}+26x^{3}+5x^{2}+12x+3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dac13eb369997aeef59e855cc798722736e3fb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:211.049ex; height:3.176ex;" alt="{\displaystyle P(x)Q(x)_{}^{}=(2x_{}^{3}+4x+1)(5x^{2}+3)=(2x_{}^{3}+4x+1)(5x^{2})+(2x^{3}+4x+1)(3)=(2x_{}^{3}\cdot 5x^{2}+4x\cdot 5x^{2}+1\cdot 5x^{2})+(2x^{3}\cdot 3+4x\cdot 3+1\cdot 3)=10x_{}^{5}+20x^{3}+5x^{2}+6x^{3}+12x+3=10x_{}^{5}+26x^{3}+5x^{2}+12x+3}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Identitate_nabarmenak">Identitate nabarmenak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=15" title="Aldatu atal hau: «Identitate nabarmenak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=15" title="Edit section's source code: Identitate nabarmenak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Identitate_(matematika)" title="Identitate (matematika)">Identitate (matematika)</a>»</div> <ul><li><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+a)^{2}=x^{2}+2ax+a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+a)^{2}=x^{2}+2ax+a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cef6184ce4b0c5a9094f93796cb6c9b31fc5bc26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.433ex; height:3.176ex;" alt="{\displaystyle (x+a)^{2}=x^{2}+2ax+a^{2}}"></span></li></ul> <ul><li><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-a)^{2}=x^{2}-2ax+a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)^{2}=x^{2}-2ax+a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39aa92f6505f4cc5ddcf8055e9ccc0c363cb6b5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.433ex; height:3.176ex;" alt="{\displaystyle (x-a)^{2}=x^{2}-2ax+a^{2}}"></span></li></ul> <ul><li><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+a)(x-a)=x^{2}-a^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+a)(x-a)=x^{2}-a^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6cbc909776c0d9a87c2302be90fa4a8e72354a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.025ex; height:3.176ex;" alt="{\displaystyle (x+a)(x-a)=x^{2}-a^{2}}"></span></li></ul> <p>Adibideak: </p> <ul><li><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+3)^{2}=x^{2}+6x+9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>3</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x+3)^{2}=x^{2}+6x+9}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91be7aa0e5b8ba008b33fd049b46ccd712c776ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.014ex; height:3.176ex;" alt="{\displaystyle (x+3)^{2}=x^{2}+6x+9}"></span></li> <li><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-1)^{2}=4x^{2}-4x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2x-1)^{2}=4x^{2}-4x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2039c01d2ebb91e50db5b52feba2ce7447488436" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.339ex; height:3.176ex;" alt="{\displaystyle (2x-1)^{2}=4x^{2}-4x+1}"></span></li> <li><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+5)\cdot (3x-5)=9x^{2}-25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>9</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (3x+5)\cdot (3x-5)=9x^{2}-25}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f445853672921f6a5fc6f480d8d1243085812f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.098ex; height:3.176ex;" alt="{\displaystyle (3x+5)\cdot (3x-5)=9x^{2}-25}"></span></li></ul> <ul><li><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}+10x+25=(x+5)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>+</mo> <mn>25</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+10x+25=(x+5)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0340030a67208fa79837fbdc036e5a6fa29e2570" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.339ex; height:3.176ex;" alt="{\displaystyle x^{2}+10x+25=(x+5)^{2}}"></span></li></ul> <ul><li><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 36x^{2}-12x+1=(6x-1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>36</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>12</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 36x^{2}-12x+1=(6x-1)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f208a1095aeeab38420b0b2e4c13d56c6fd7eda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.664ex; height:3.176ex;" alt="{\displaystyle 36x^{2}-12x+1=(6x-1)^{2}}"></span></li> <li><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 4x^{2}-1=(2x+1)\cdot (2x-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}-1=(2x+1)\cdot (2x-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a956fd2fa2513afac8f3546c691747438d16e05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.935ex; height:3.176ex;" alt="{\displaystyle 4x^{2}-1=(2x+1)\cdot (2x-1)}"></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="Zatiketa">Zatiketa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=16" title="Aldatu atal hau: «Zatiketa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=16" title="Edit section's source code: Zatiketa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Monomioen koefizienteak zatituz eta indeterminatu berdinen mailak kenduz lortzen da. Adibidez, </p> <ul><li><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 {42x_{}^{3} \over 6x}=({42 \over 6})\cdot x_{}^{3-1}=7x_{}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>42</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> <mrow> <mn>6</mn> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>42</mn> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>=</mo> <mn>7</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {42x_{}^{3} \over 6x}=({42 \over 6})\cdot x_{}^{3-1}=7x_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8716c665200cddf84a71c4d0647fb90c0a179b60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.422ex; height:5.676ex;" alt="{\displaystyle {42x_{}^{3} \over 6x}=({42 \over 6})\cdot x_{}^{3-1}=7x_{}^{2}}"></span></li> <li><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 {15x_{}^{3}y_{}^{5} \over 10xy_{}^{3}}=({15 \over 10})\cdot x_{}^{3-1}\cdot y_{}^{5-3}={3 \over 2}x_{}^{2}y_{}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>15</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mrow> <mrow> <mn>10</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>15</mn> <mn>10</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>3</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {15x_{}^{3}y_{}^{5} \over 10xy_{}^{3}}=({15 \over 10})\cdot x_{}^{3-1}\cdot y_{}^{5-3}={3 \over 2}x_{}^{2}y_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfe69c678ba0167da8e81d437cf07e0bda1c769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.682ex; height:6.343ex;" alt="{\displaystyle {15x_{}^{3}y_{}^{5} \over 10xy_{}^{3}}=({15 \over 10})\cdot x_{}^{3-1}\cdot y_{}^{5-3}={3 \over 2}x_{}^{2}y_{}^{2}}"></span></li> <li><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 {11x_{}^{3}y_{}^{5}z_{}^{2} \over 22xy_{}^{3}z_{}^{2}}=({11 \over 22})\cdot x_{}^{3-1}\cdot y_{}^{5-3}\cdot z_{}^{2-2}={1 \over 2}x_{}^{2}y_{}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>11</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mrow> <mn>22</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>11</mn> <mn>22</mn> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>⋅</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> <mo>−</mo> <mn>3</mn> </mrow> </msubsup> <mo>⋅</mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {11x_{}^{3}y_{}^{5}z_{}^{2} \over 22xy_{}^{3}z_{}^{2}}=({11 \over 22})\cdot x_{}^{3-1}\cdot y_{}^{5-3}\cdot z_{}^{2-2}={1 \over 2}x_{}^{2}y_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6af87ff5499c4842fc42529cdefc2ef297b0028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.751ex; height:6.343ex;" alt="{\displaystyle {11x_{}^{3}y_{}^{5}z_{}^{2} \over 22xy_{}^{3}z_{}^{2}}=({11 \over 22})\cdot x_{}^{3-1}\cdot y_{}^{5-3}\cdot z_{}^{2-2}={1 \over 2}x_{}^{2}y_{}^{2}}"></span></li></ul><p> Zenbaki errealekin egindako zatiketak polinomioekin egitekotan, zatikizunaren mailak zatitzailearen maila baino handiagoa edo berdina izan beharko du. Kasu horretan, zatiketa egiten ikasteko adibide honi jarraituko diogu: <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)=63x_{}^{3}-84x_{}^{2}+3x+20}"> <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> <mn>63</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>84</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=63x_{}^{3}-84x_{}^{2}+3x+20}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720af05000ab2f520d0b74c2229deb95a9a65a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.739ex; height:3.176ex;" alt="{\displaystyle P(x)=63x_{}^{3}-84x_{}^{2}+3x+20}"></span> eta <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> polinomioak izanda, <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)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {P(x)}:{Q(x)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7128b93215fdd63d530f5c1ba5083d984a10e31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.799ex; height:2.843ex;" alt="{\displaystyle {P(x)}:{Q(x)}}"></span> lortzeko:</p><ul class="gallery mw-gallery-packed"> <li class="gallerybox" style="width: 299.33333333333px"> <div class="thumb" style="width: 297.33333333333px;"><span typeof="mw:File"><a href="/wiki/Fitxategi:Zatiketa_1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/e/e2/Zatiketa_1.jpg" decoding="async" width="298" height="120" class="mw-file-element" data-file-width="329" data-file-height="133" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 282px"> <div class="thumb" style="width: 280px;"><span typeof="mw:File"><a href="/wiki/Fitxategi:Polinomioaren_zatiketa_2.pauso_.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/38/Polinomioaren_zatiketa_2.pauso_.jpg/420px-Polinomioaren_zatiketa_2.pauso_.jpg" decoding="async" width="280" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/38/Polinomioaren_zatiketa_2.pauso_.jpg 1.5x" data-file-width="490" data-file-height="210" /></a></span></div> <div class="gallerytext"></div> </li> <li class="gallerybox" style="width: 254px"> <div class="thumb" style="width: 252px;"><span typeof="mw:File"><a href="/wiki/Fitxategi:Polinomioaren_zatiketa_3.pauso_.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Polinomioaren_zatiketa_3.pauso_.jpg/378px-Polinomioaren_zatiketa_3.pauso_.jpg" decoding="async" width="252" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/09/Polinomioaren_zatiketa_3.pauso_.jpg/567px-Polinomioaren_zatiketa_3.pauso_.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/0/09/Polinomioaren_zatiketa_3.pauso_.jpg 2x" data-file-width="583" data-file-height="278" /></a></span></div> <div class="gallerytext"></div> </li> </ul> <div class="mw-heading mw-heading2"><h2 id="Ariketak"><figure class="mw-halign-left" typeof="mw:File"><a href="/wiki/Fitxategi:Jakindun_logoa.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/24px-Jakindun_logoa.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/36px-Jakindun_logoa.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Jakindun_logoa.png/48px-Jakindun_logoa.png 2x" data-file-width="225" data-file-height="225" /></a><figcaption></figcaption></figure> Ariketak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=17" title="Aldatu atal hau: «Ariketak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=17" title="Edit section's source code: Ariketak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul class="gallery mw-gallery-packed" style="background-color: #fef6e7; margin-left: 0;"> <li class="gallerycaption">Monomioak eta polinomioak</li> <li class="gallerybox" style="width: 215.33333333333px"> <div class="thumb" style="width: 213.33333333333px;"><span typeof="mw:File"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/320px-seek%3D3-Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="214" height="120" data-durationhint="762" data-mwtitle="Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.720p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="720p.vp9.webm" data-width="1280" data-height="720" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.1080p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="1080p.vp9.webm" data-width="1920" data-height="1080" /><source src="//upload.wikimedia.org/wikipedia/commons/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm" type="video/webm; codecs="vp9, opus"" data-width="3840" data-height="2160" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/8/8e/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm/Monomio_eta_polinomioen_arteko_zatiketak_azalpena_ariketak.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span></span></div> <div class="gallerytext"><b>Monomioen</b> eta <b>polinomioen</b> arteko zatiketak.</div> </li> <li class="gallerybox" style="width: 215.33333333333px"> <div class="thumb" style="width: 213.33333333333px;"><span typeof="mw:File"><span><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Deskonposizio_polinomikoa.webm/320px-seek%3D3-Deskonposizio_polinomikoa.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="214" height="120" data-durationhint="240" data-mwtitle="Deskonposizio_polinomikoa.webm" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Deskonposizio_polinomikoa.webm/Deskonposizio_polinomikoa.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="480" /><source src="//upload.wikimedia.org/wikipedia/commons/f/f3/Deskonposizio_polinomikoa.webm" type="video/webm; codecs="vp8, vorbis"" data-width="1077" data-height="606" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Deskonposizio_polinomikoa.webm/Deskonposizio_polinomikoa.webm.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Deskonposizio_polinomikoa.webm/Deskonposizio_polinomikoa.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Deskonposizio_polinomikoa.webm/Deskonposizio_polinomikoa.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/f/f3/Deskonposizio_polinomikoa.webm/Deskonposizio_polinomikoa.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span></span></div> <div class="gallerytext"><b>Deskonposizio polinomikoa</b> lantzeko ariketa.</div> </li> </ul> <div class="mw-heading mw-heading4"><h4 id="Ruffiniren_erregela">Ruffiniren erregela</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=18" title="Aldatu atal hau: «Ruffiniren erregela»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=18" title="Edit section's source code: Ruffiniren erregela"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Ruffiniren_erregela" title="Ruffiniren erregela">Ruffiniren erregela</a>»</div> <p>Zatiketa baten zatitzailea (x+r) edo (x-r) erakoa bada, orduan zatiketa Ruffiniren bidez egin ahal dugu. </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 P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +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> <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}+\cdots +a_{1}x+a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbede04a95764c2dccd8ceda314b15f9caf62c29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.159ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}}"></span>zatikizun eta <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-r\,\!}"> <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> <mi>r</mi> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=x-r\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbf983a63b7b6d53b45532f7d6328cb3aa7ead9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:13.682ex; height:2.843ex;" alt="{\displaystyle Q(x)=x-r\,\!}"></span>zatitzaile izanda,urrats hauei jarraituko diegu: </p><p><b>1.</b> <i>P</i>(<i>x</i>) polinomioaren koefizienteak ordenaturik idatzi behar dira. Eta ondoren, lerro bat beherago, zatitzailea den <i>x-r</i> binomioko <i>r</i> jarri behar da, irudiko marra laguntzaileekin batera: </p> <pre> | a<sub>n</sub> a<sub>n-1</sub> ... a<sub>1</sub> a<sub>0</sub> | r | ----|--------------------------------------------------------- | | </pre> <p><b>2.</b> Ezkerreko lehenengo koefizientea behera eraman, hura aldatu gabe: </p> <pre> | a<sub>n</sub> a<sub>n-1</sub> ... a<sub>1</sub> a<sub>0</sub> | r | ----|--------------------------------------------------------- | a<sub>n</sub>= | | b<sub>n-1</sub> | </pre> <p><b>3.</b> Behera pasatutako koefiziente hori r balioaz biderkatu eta polinomioaren hurrengo koefizientearen azpian jarri: </p> <pre> | a<sub>n</sub> a<sub>n-1</sub> ... a<sub>1</sub> a<sub>0</sub> | r | b<sub>n-1</sub>r ----|--------------------------------------------------------- | a<sub>n</sub> | | = b<sub>n-1</sub> | </pre> <p><b>4.</b> Zutabe bereko bi balio hauen batuketa egin: </p> <pre> | a<sub>n</sub> a<sub>n-1</sub> ... a<sub>1</sub> a<sub>0</sub> | r | b<sub>n-1</sub>r ----|--------------------------------------------------------- | a<sub>n</sub> a<sub>n-1</sub>+(b<sub>n-1</sub>r) | | = b<sub>n-1</sub> = b<sub>n-2</sub> | </pre> <p><b>5.</b> 3. eta 4. pausoak errepikatu lerroa bukatu arte: </p> <pre> | a<sub>n</sub> a<sub>n-1</sub> ... a<sub>1</sub> a<sub>0</sub> | r | b<sub>n-1</sub>r ... b<sub>1</sub>r b<sub>0</sub>r ----|--------------------------------------------------------- | a<sub>n</sub> a<sub>n-1</sub>+(b<sub>n-1</sub>r) ... a<sub>1</sub>+b<sub>1</sub>r a<sub>0</sub>+b<sub>0</sub>r | | = b<sub>n-1</sub> = b<sub>n-2</sub> ... = b<sub>0</sub> = s </pre> <p>Adibidez: </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 P(x)=2x^{3}+3x^{2}-4\,\!}"> <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> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x^{3}+3x^{2}-4\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc0ce525a8d23fa9810100448b0a7f0654b07aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:22.306ex; height:3.176ex;" alt="{\displaystyle P(x)=2x^{3}+3x^{2}-4\,\!}"></span> </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 Q(x)=x+1=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> <mo>=</mo> <mi>x</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=x+1=x-(-1)\,\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e996e5798ca99b36af8a9e26ee8d9995cbdd9ba7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:25.844ex; height:2.843ex;" alt="{\displaystyle Q(x)=x+1=x-(-1)\,\!}"></span></dd></dl> <p>Ohartu behar da <i>x+1</i> binomioa x-(-1) bihurtzen dela, <i>x-r</i> erakoa izateko: </p><p><i></i> </p><p><b>1.</b> </p><p>Koefizienteak bere lekuan jarri: </p> <pre> | 2 3 0 -4 | -1 | ----|---------------------------- | | </pre> <p>Ohartu behar da polinomioan <i>x</i> terminoaren koefizientea 0 dela. </p><p><b>2.</b> </p><p>Lehenengo koefizientea behera eraman: </p> <pre> | 2 3 0 -4 | -1 | ----|---------------------------- | 2 | </pre> <p><b>3.</b> </p> <pre>-1×2=-2 egin | 2 3 0 -4 | -1 | -2 ----|---------------------------- | 2 | </pre> <p><b>4.</b> </p><p>3-2=1 </p> <pre> | 2 3 0 -4 | -1 | -2 ----|---------------------------- | 2 1 | </pre> <p><b>5.</b> </p><p>Lerroa bukatu arte jarraituz: </p> <pre> | 2 3 0 -4 | -1 | -2 -1 1 ----|------------------------------- | 2 1 -1 -3 |{zatidura koefizienteak}{hondarra} </pre> <p>Beraz: </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 {\frac {2x^{3}+3x^{2}-4}{x+1}}=2x^{2}+x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2x^{3}+3x^{2}-4}{x+1}}=2x^{2}+x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c56466354785e30bc0d380e733e476309c6d0453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:29.59ex; height:5.843ex;" alt="{\displaystyle {\frac {2x^{3}+3x^{2}-4}{x+1}}=2x^{2}+x-1}"></span> da eta hondarra -3</dd> <dd></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Faktore_komuna_ateratzea">Faktore komuna ateratzea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=19" title="Aldatu atal hau: «Faktore komuna ateratzea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=19" title="Edit section's source code: Faktore komuna ateratzea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="font-style: italic; padding-left: 2em; margin-bottom: 0.5em;">Sakontzeko, irakurri: «<a href="/wiki/Faktorizazio" title="Faktorizazio">Faktorizazio</a>»</div> <p>Polinomio batean faktore komuna atera ahal izateko, faktore hori batugai guztietan egon behar da<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>. </p> <ul><li><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)=2x_{}^{4}+4x_{}^{2}-6x=2x\cdot (x_{}^{3}+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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>4</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{4}+4x_{}^{2}-6x=2x\cdot (x_{}^{3}+2x-3)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba2fa694b28d5c3c39f033ffb8ae8647f97417e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.047ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{4}+4x_{}^{2}-6x=2x\cdot (x_{}^{3}+2x-3)}"></span></li> <li><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)=12x_{}^{2}y-15xy_{}^{2}+3xy=3xy\cdot (4x-5y+1)}"> <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> <mn>12</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mi>y</mi> <mo>−</mo> <mn>15</mn> <mi>x</mi> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mi>y</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x,y)=12x_{}^{2}y-15xy_{}^{2}+3xy=3xy\cdot (4x-5y+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/406183a83252e4e26ba51ca190a8421577444767" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.122ex; height:3.176ex;" alt="{\displaystyle P(x,y)=12x_{}^{2}y-15xy_{}^{2}+3xy=3xy\cdot (4x-5y+1)}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Polinomioen_erroak">Polinomioen erroak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=20" title="Aldatu atal hau: «Polinomioen erroak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=20" title="Edit section's source code: Polinomioen erroak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomio baten erroak P(x)=0 ekuazioaren soluzioak dira<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>. Beraz, <i>a</i> zenbaki bati P(x) polinomioaren erroa esaten zaio baldin eta P(a)=0 bada. Adibidez: </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 P(x)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{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> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40e9af1a8d8b7fe115dbc41346f027402b8f1f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.301ex; height:3.176ex;" alt="{\displaystyle P(x)=2x_{}^{7}+3x_{}^{6}+x_{}^{5}-6x_{}^{4}+2x_{}^{3}-5x_{}^{2}}"></span> polinomioa izanda, </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 P(0)=2\cdot 0_{}^{7}+3\cdot 0_{}^{6}+0_{}^{5}-6\cdot 0_{}^{4}+2\cdot 0_{}^{3}-5\cdot 0_{}^{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <mo>⋅</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <msubsup> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(0)=2\cdot 0_{}^{7}+3\cdot 0_{}^{6}+0_{}^{5}-6\cdot 0_{}^{4}+2\cdot 0_{}^{3}-5\cdot 0_{}^{2}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c702e752f328ade391dd84bc6896a173b7532b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.787ex; height:3.176ex;" alt="{\displaystyle P(0)=2\cdot 0_{}^{7}+3\cdot 0_{}^{6}+0_{}^{5}-6\cdot 0_{}^{4}+2\cdot 0_{}^{3}-5\cdot 0_{}^{2}=0}"></span> eta <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(1)=2\cdot 1_{}^{7}+3\cdot 1_{}^{6}+1_{}^{5}-6\cdot 1_{}^{4}+2\cdot 1_{}^{3}-5\cdot 1_{}^{2}=-3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mo>⋅</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msubsup> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> <mo>−</mo> <mn>6</mn> <mo>⋅</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mo>−</mo> <mn>5</mn> <mo>⋅</mo> <msubsup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(1)=2\cdot 1_{}^{7}+3\cdot 1_{}^{6}+1_{}^{5}-6\cdot 1_{}^{4}+2\cdot 1_{}^{3}-5\cdot 1_{}^{2}=-3}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c3881d495c5cd5393a237006e682bbd0c805a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.595ex; height:3.176ex;" alt="{\displaystyle P(1)=2\cdot 1_{}^{7}+3\cdot 1_{}^{6}+1_{}^{5}-6\cdot 1_{}^{4}+2\cdot 1_{}^{3}-5\cdot 1_{}^{2}=-3}"></span>; </p><p>hori dela eta, x=0 polinomioaren erroa da eta x=1 ez. </p><p>M mailako polinomio batean, gehienez M erro aurki ditzakegu. Erroak berdinak edo desberdinak izan daitezke. Erro bat behin agertzen denean, erro sinple deritzo; erroa behin baino gehiagotan agertzen denean, aldiz, izen hauek jasotzen ditu erroak: erro bikoitza (bitan agertzen bada), hirukoitza (hiru alditan agertzen bada)... </p> <div class="mw-heading mw-heading2"><h2 id="Polinomioen_faktorizazioa">Polinomioen faktorizazioa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=21" title="Aldatu atal hau: «Polinomioen faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=21" title="Edit section's source code: Polinomioen faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Zenbakiak faktorizatu daitezkeen bezala, polinomioak ere faktorizatu daitezke, "oinarrizko" polinomio batzuen biderkadura modura idatziz. Hala ere, badaude polinomio batzuk (zenbakien kasuan "zenbaki lehenak"), ezin daitezkeenak faktorizatu eta horiei, irreduzibleak deritze. </p> <div class="mw-heading mw-heading3"><h3 id="Kronecker_metodoa">Kronecker metodoa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=22" title="Aldatu atal hau: «Kronecker metodoa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=22" title="Edit section's source code: Kronecker metodoa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomio osoek faktore polinomial osoetan faktorizatu behar direnez, eta balio osoen polinomio osoen ebaluazioak zenbaki osoak sortu behar dituztenez, polinomio baten balio osoak, kopuru finituan soilik hartu behar dira kontuan, eta ondorioz, faktore polinomiko posibleen kopuru mugatu bat baino ez dute sortzen. </p><p>Adibidez, kontsideratu dezagun hau: </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 f(x)=x^{5}+x^{4}+x^{3}+x^{2}+x+2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <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 f(x)=x^{5}+x^{4}+x^{3}+x^{2}+x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b766dc545f35286124aa785a9ab1e3542ffff48f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.746ex; height:3.176ex;" alt="{\displaystyle f(x)=x^{5}+x^{4}+x^{3}+x^{2}+x+2}"></span> </p><p>Faktore polinomiko hauek <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> gorputzaren gainean badaude, orduan gutxienez batek bigarren mailakoa edo bajuagokoa izan beharra du. 3 balio besterik ez dira behar bigarren mailako polinomioa aurkitzeko. Beraz, <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 f(0)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(0)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce2b15285ebb5da7e738340aec574febcc645d7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(0)=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 f(1)=6,f(-1)=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>6</mn> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=6,f(-1)=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/602f0836e82686e50cc984264a8f14dea068399d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.865ex; height:2.843ex;" alt="{\displaystyle f(1)=6,f(-1)=2}"></span>. Kontuan izan, hauetako balioren batek 0 ematen badu, lortu dugula polinomio honen erro bat (eta beraz faktore bat). Baina aukeratutakoek ez badute 0 ematen emaitza bezala, emaitza bakoitzak zatitzaile kopuru finitu bat du. Adibidez 2 zenbakia, horrela faktoriza dezakegu: </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 1\times 2,2\times 1,(-1)\times (-2),(-2)\times (-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>×</mo> <mn>1</mn> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times 2,2\times 1,(-1)\times (-2),(-2)\times (-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dda1c4cdc9efb612483f4f2f1b0570918af0dc1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.233ex; height:2.843ex;" alt="{\displaystyle 1\times 2,2\times 1,(-1)\times (-2),(-2)\times (-1)}"></span> </p><p>Hortaz, bigarren mailako faktore polinomiko bat existituz gero, balioak hauexek hartu ditzake: </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 1,2,(-1),(-2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,(-1),(-2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e617165b15290dcc35619ed2daa640a48d8298" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.987ex; height:2.843ex;" alt="{\displaystyle 1,2,(-1),(-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 x=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/953917eaf52f2e1baad54c8c9e3d6f9bb3710cdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x=0}"></span> eta <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"> <mi>x</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </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/4fefa55268918f98da2e0dcc19ea86d78f84ac56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.399ex; height:2.343ex;" alt="{\displaystyle x=-1}"></span> -en kasuan. </p><p>Zortzi modu desberdin daude 6 zenbakia faktorizatzeko (modu bat duen zatitzaile bakoitzarekiko), orduan </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 4\times 4\times 8=148}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>8</mn> <mo>=</mo> <mn>148</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4\times 4\times 8=148}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb2ae8e711a9f3e0f57ab4448560bba35a3b28f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.754ex; height:2.176ex;" alt="{\displaystyle 4\times 4\times 8=148}"></span> konbinazio posible daude. Erdiak negatiboak direnez konbinazio posibleetatik kendu ditzakegu, eta beraz <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 64}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>64</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 64}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482f6722cbd449a0df54e03c71143afc7cb1ea4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle 64}"></span> konbinazio probatzea geratzen zaigu bigarren mailako polinomio zuzena aurkitu arte. </p><p>Probak egin ondoren konklusio honetara iristen gara, <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> bigarren mailako polinomioen faktorizazio modu bakarra dagoela <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>-n, eta hauxe da: </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 p(x)=x^{2}+x+1}"> <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>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=x^{2}+x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86795e4491a6ebc874d8ccbf6fa2d95f89f139a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:18.053ex; height:3.176ex;" alt="{\displaystyle p(x)=x^{2}+x+1}"></span> emaitza hauetatik eraikia <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(0)=1,p(1)=3,p(-1)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(0)=1,p(1)=3,p(-1)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c4c98479c2588a4d35ca0270b91298da819087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:29.172ex; height:2.843ex;" alt="{\displaystyle p(0)=1,p(1)=3,p(-1)=1}"></span>. </p><p>Azkenik, <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(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 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/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span> -ren gatik zatituz beste polinomioa lortzen dugu <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^{3}-x+2}"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)=x^{3}-x+2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1b3bc3bc05c1041acc903fa53d8264bf8873109" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.864ex; height:3.176ex;" alt="{\displaystyle q(x)=x^{3}-x+2}"></span> eta ondorioz, <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 f(x)=p(x)\times q(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=p(x)\times q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26f6a55a22abbfaf2295d46cc4758b7fc7adaf71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.873ex; height:2.843ex;" alt="{\displaystyle f(x)=p(x)\times q(x)}"></span>. Eta orain, <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/8cb7afced134ef75572e5314a5d278c2d644f438" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle p(x)}"></span> eta <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)}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c38bbafe34a043d284f19231b946a76c0a4b16b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.209ex; height:2.843ex;" alt="{\displaystyle q(x)}"></span>-rentzat hasi gaitezke bilatzen faktoreak, baina kasualitatez bi polinomio hauek irreduzibleak direla <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 \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span> gorputzean, orduan <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 f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}"></span> -ren faktorizazio irreduziblea hauxe da: </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 f(x)=p(x)\times q(x)=(x^{2}+x+1)\times (x^{3}-x+2)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> <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>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=p(x)\times q(x)=(x^{2}+x+1)\times (x^{3}-x+2)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940c92342c60570498cff516e559a7a38ed3903b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.545ex; height:3.176ex;" alt="{\displaystyle f(x)=p(x)\times q(x)=(x^{2}+x+1)\times (x^{3}-x+2)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Erreferentziak">Erreferentziak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=23" title="Aldatu atal hau: «Erreferentziak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=23" title="Edit section's source code: Erreferentziak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"></span>.</span> </li> <li id="cite_note-2"><a href="#cite_ref-2">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="https://sites.google.com/a/burdinibarrabhi.net/aljebra/identitate-nabarmenak/atera-faktore">«Atera faktore komuna - Aljebra»</a> <i>sites.google.com</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2018-11-21)</small></span></span>.</span> </li> <li id="cite_note-3"><a href="#cite_ref-3">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation"><a rel="nofollow" class="external text" href="https://sites.google.com/a/lekeitiobhi.net/dbh-4ko-buruketak/6-10-buruketak/5---polinomio-baten-erroak">«5.- Polinomio baten erroak - MATEMATIKA 4.DBH FLIPPED»</a> <i>sites.google.com</i> <span class="reference-accessdate"><small>(Noiz kontsultatua: 2018-11-21)</small></span></span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Kanpo_estekak">Kanpo estekak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Polinomio&veaction=edit&section=24" title="Aldatu atal hau: «Kanpo estekak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Polinomio&action=edit&section=24" title="Edit section's source code: Kanpo estekak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="clear:both;"></div><style data-mw-deduplicate="TemplateStyles:r7786466">.mw-parser-output .mw-authority-control{margin-top:1.5em}.mw-parser-output .mw-authority-control .navbox hr:last-child{display:none}.mw-parser-output .mw-authority-control .navbox+.mw-mf-linked-projects{display:none}.mw-parser-output .mw-authority-control,.mw-parser-output .mw-mf-linked-projects{border:1px solid #a2a9b1;font-size:88%}.mw-parser-output .mw-authority-control .mw-mf-linked-projects ul li{margin-bottom:0}</style><div class="mw-authority-control"><div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r9236167">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px}.mw-parser-output .navbox 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title="wikidata:Q43260">Q43260</a></span></li> <li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Commonscat"><img alt="Commonscat" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/15px-Commons-logo.svg.png" decoding="async" width="15" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/23px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Multimedia:</span> <span class="uid"><span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Polynomials">Polynomials</a></span> / <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Special:MediaSearch?type=image&search=%22Q43260%22">Q43260</a></span></span></li></ul> <hr /> 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<li><span style="white-space:nowrap;"><a href="/wiki/National_Library_of_the_Czech_Republic" class="mw-redirect" title="National Library of the Czech Republic">NKC</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph135425">ph135425</a></span></li> <li><b>Hiztegiak eta entziklopediak</b></li> <li><span style="white-space:nowrap;"><a href="/wiki/Encyclop%C3%A6dia_Britannica" title="Encyclopædia Britannica">Britannica</a>:</span> <span class="uid"><a rel="nofollow" class="external text" href="https://www.britannica.com/topic/polynomial">url</a></span></li></ul> </div></td></tr></tbody></table></div><div class="mw-mf-linked-projects hlist"> <ul><li><span style="white-space:nowrap;"><span typeof="mw:File"><a href="/wiki/Wikidata" title="Wikidata"><img alt="Wd" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/ff/Wikidata-logo.svg/20px-Wikidata-logo.svg.png" decoding="async" width="20" height="11" 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