Faktorizazio - 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=Faktorizazio" title="Izen ematera gonbidatzen zaitugu. 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[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-Zenbaki_osoen_faktorizazioa" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zenbaki_osoen_faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Zenbaki osoen faktorizazioa</span> </div> </a> <button aria-controls="toc-Zenbaki_osoen_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 Zenbaki osoen faktorizazioa azpiatal</span> </button> <ul id="toc-Zenbaki_osoen_faktorizazioa-sublist" class="vector-toc-list"> <li id="toc-Adibidea" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Adibidea"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Adibidea</span> </div> </a> <ul id="toc-Adibidea-sublist" class="vector-toc-list"> </ul> </li> </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">2</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-Faktorizazioaren_historia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Faktorizazioaren_historia"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Faktorizazioaren historia</span> </div> </a> <ul id="toc-Faktorizazioaren_historia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metodo_orokorrak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metodo_orokorrak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Metodo orokorrak</span> </div> </a> <ul id="toc-Metodo_orokorrak-sublist" class="vector-toc-list"> <li id="toc-Faktore_komuna" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Faktore_komuna"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Faktore komuna</span> </div> </a> <ul id="toc-Faktore_komuna-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multzokatze_bidezko_faktore_komuna" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Multzokatze_bidezko_faktore_komuna"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Multzokatze bidezko faktore komuna</span> </div> </a> <ul id="toc-Multzokatze_bidezko_faktore_komuna-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Faktorearen_teorema" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Faktorearen_teorema"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Faktorearen teorema</span> </div> </a> <ul id="toc-Faktorearen_teorema-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Aldagai_baten_kasua,_erroen_propietateak_erabiliz" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Aldagai_baten_kasua,_erroen_propietateak_erabiliz"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>Aldagai baten kasua, erroen propietateak erabiliz</span> </div> </a> <ul id="toc-Aldagai_baten_kasua,_erroen_propietateak_erabiliz-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Patroi_ezagunak" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Patroi_ezagunak"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Patroi ezagunak</span> </div> </a> <ul id="toc-Patroi_ezagunak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polinomioaren_erroen_formula" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polinomioaren_erroen_formula"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Polinomioaren erroen formula</span> </div> </a> <ul id="toc-Polinomioaren_erroen_formula-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zenbaki_konplexuen_gaineko_faktorizazioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zenbaki_konplexuen_gaineko_faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Zenbaki konplexuen gaineko faktorizazioa</span> </div> </a> <ul id="toc-Zenbaki_konplexuen_gaineko_faktorizazioa-sublist" class="vector-toc-list"> <li id="toc-Bi_karratuen_batura" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Bi_karratuen_batura"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5.1</span> <span>Bi karratuen batura</span> </div> </a> <ul id="toc-Bi_karratuen_batura-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Matrizeak" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrizeak"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Matrizeak</span> </div> </a> <button aria-controls="toc-Matrizeak-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 Matrizeak azpiatal</span> </button> <ul id="toc-Matrizeak-sublist" class="vector-toc-list"> <li id="toc-LU_Faktorizazioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#LU_Faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>LU Faktorizazioa</span> </div> </a> <ul id="toc-LU_Faktorizazioa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-'"`UNIQ--postMath-0000002B-QINU`"'_Faktorizazioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#'"`UNIQ--postMath-0000002B-QINU`"'_Faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>'"`UNIQ--postMath-0000002B-QINU`"' Faktorizazioa</span> </div> </a> <ul id="toc-'"`UNIQ--postMath-0000002B-QINU`"'_Faktorizazioa-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-QR_Faktorizazioa" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#QR_Faktorizazioa"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>QR Faktorizazioa</span> </div> </a> <ul id="toc-QR_Faktorizazioa-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">4</span> <span>Erreferentziak</span> </div> </a> <ul id="toc-Erreferentziak-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografia" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliografia"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Bibliografia</span> </div> </a> <ul id="toc-Bibliografia-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">6</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">Faktorizazio</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. 62 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-62" 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">62 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/Faktorisasie" title="Faktorisasie – afrikaansa" lang="af" hreflang="af" data-title="Faktorisasie" 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/%D8%AA%D8%AD%D9%84%D9%8A%D9%84_%D8%A5%D9%84%D9%89_%D8%B9%D9%88%D8%A7%D9%85%D9%84" title="تحليل إلى عوامل – arabiera" lang="ar" hreflang="ar" data-title="تحليل إلى عوامل" data-language-autonym="العربية" data-language-local-name="arabiera" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%89%E0%A7%8E%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A6%95_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="উৎপাদক বিশ্লেষণ – assamera" lang="as" hreflang="as" data-title="উৎপাদক বিশ্লেষণ" data-language-autonym="অসমীয়া" data-language-local-name="assamera" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Faktorizasiya" title="Faktorizasiya – azerbaijanera" lang="az" hreflang="az" data-title="Faktorizasiya" data-language-autonym="Azərbaycanca" data-language-local-name="azerbaijanera" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F" title="Факторизация – bulgariera" lang="bg" hreflang="bg" data-title="Факторизация" 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%89%E0%A7%8E%E0%A6%AA%E0%A6%BE%E0%A6%A6%E0%A6%95%E0%A7%87_%E0%A6%AC%E0%A6%BF%E0%A6%B6%E0%A7%8D%E0%A6%B2%E0%A7%87%E0%A6%B7%E0%A6%A3" title="উৎপাদকে বিশ্লেষণ – bengalera" lang="bn" hreflang="bn" data-title="উৎপাদকে বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="bengalera" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factoritzaci%C3%B3" title="Factorització – katalana" lang="ca" hreflang="ca" data-title="Factorització" 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%BE%D8%A7%D9%88%D8%A6%DB%95%D9%86%D8%AC%D8%A7%D9%85_%D9%88_%D8%B4%DB%8C%D8%AA%DB%95%DA%B5%DA%A9%D8%B1%D8%AF%D9%86" title="ھاوئەنجام و شیتەڵکردن – erdialdeko kurduera" lang="ckb" hreflang="ckb" data-title="ھاوئەنجام و شیتەڵکردن" 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/Faktorizace" title="Faktorizace – txekiera" lang="cs" hreflang="cs" data-title="Faktorizace" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8" title="Факторизаци – txuvaxera" lang="cv" hreflang="cv" data-title="Факторизаци" 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/Ffactorau_cysefin" title="Ffactorau cysefin – galesa" lang="cy" hreflang="cy" data-title="Ffactorau cysefin" 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/Faktorisering" title="Faktorisering – daniera" lang="da" hreflang="da" data-title="Faktorisering" 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/Faktorisierung" title="Faktorisierung – alemana" lang="de" hreflang="de" data-title="Faktorisierung" data-language-autonym="Deutsch" data-language-local-name="alemana" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%B1%CE%B3%CE%BF%CE%BD%CF%84%CE%BF%CF%80%CE%BF%CE%AF%CE%B7%CF%83%CE%B7" title="Παραγοντοποίηση – greziera" lang="el" hreflang="el" data-title="Παραγοντοποίηση" 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/Factorization" title="Factorization – ingelesa" lang="en" hreflang="en" data-title="Factorization" 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/Faktorado" title="Faktorado – esperantoa" lang="eo" hreflang="eo" data-title="Faktorado" 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/Factorizaci%C3%B3n" title="Factorización – gaztelania" lang="es" hreflang="es" data-title="Factorización" data-language-autonym="Español" data-language-local-name="gaztelania" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%D8%AA%D8%AD%D8%A7%D8%AF_%D9%88_%D8%AA%D8%AC%D8%B2%DB%8C%D9%87" title="اتحاد و تجزیه – persiera" lang="fa" hreflang="fa" data-title="اتحاد و تجزیه" 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/Tekij%C3%A4" title="Tekijä – finlandiera" lang="fi" hreflang="fi" data-title="Tekijä" data-language-autonym="Suomi" data-language-local-name="finlandiera" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Valdur" title="Valdur – faroera" lang="fo" hreflang="fo" data-title="Valdur" data-language-autonym="Føroyskt" data-language-local-name="faroera" class="interlanguage-link-target"><span>Føroyskt</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Factorisation" title="Factorisation – frantsesa" lang="fr" hreflang="fr" data-title="Factorisation" data-language-autonym="Français" data-language-local-name="frantsesa" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Factorizaci%C3%B3n" title="Factorización – galiziera" lang="gl" hreflang="gl" data-title="Factorización" 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%99%D7%A8%D7%95%D7%A7_%D7%9C%D7%92%D7%95%D7%A8%D7%9E%D7%99%D7%9D" title="פירוק לגורמים – hebreera" lang="he" hreflang="he" data-title="פירוק לגורמים" 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%97%E0%A5%81%E0%A4%A3%E0%A4%A8%E0%A4%96%E0%A4%A3%E0%A5%8D%E0%A4%A1" title="गुणनखण्ड – hindia" lang="hi" hreflang="hi" data-title="गुणनखण्ड" data-language-autonym="हिन्दी" data-language-local-name="hindia" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Faktoriz%C3%A1ci%C3%B3" title="Faktorizáció – hungariera" lang="hu" hreflang="hu" data-title="Faktorizáció" data-language-autonym="Magyar" data-language-local-name="hungariera" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktorisasi" title="Faktorisasi – indonesiera" lang="id" hreflang="id" data-title="Faktorisasi" 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/Faktorigo" title="Faktorigo – idoa" lang="io" hreflang="io" data-title="Faktorigo" 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/%C3%9E%C3%A1ttun" title="Þáttun – islandiera" lang="is" hreflang="is" data-title="Þáttun" 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/Fattorizzazione" title="Fattorizzazione – italiera" lang="it" hreflang="it" data-title="Fattorizzazione" 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%9B%A0%E6%95%B0%E5%88%86%E8%A7%A3" title="因数分解 – japoniera" lang="ja" hreflang="ja" data-title="因数分解" 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-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B8%EC%88%98%EB%B6%84%ED%95%B4" title="인수분해 – koreera" lang="ko" hreflang="ko" data-title="인수분해" 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/In_factores_resolutio" title="In factores resolutio – latina" lang="la" hreflang="la" data-title="In factores resolutio" data-language-autonym="Latina" data-language-local-name="latina" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Fattorizzazion" title="Fattorizzazion – Lombard" lang="lmo" hreflang="lmo" data-title="Fattorizzazion" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Faktorizavimas" title="Faktorizavimas – lituaniera" lang="lt" hreflang="lt" data-title="Faktorizavimas" data-language-autonym="Lietuvių" data-language-local-name="lituaniera" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Факторизација – mazedoniera" lang="mk" hreflang="mk" data-title="Факторизација" data-language-autonym="Македонски" data-language-local-name="mazedoniera" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pemfaktoran" title="Pemfaktoran – malaysiera" lang="ms" hreflang="ms" data-title="Pemfaktoran" 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/Factorisatie" title="Factorisatie – nederlandera" lang="nl" hreflang="nl" data-title="Factorisatie" 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/Faktorisering" title="Faktorisering – nynorsk (norvegiera)" lang="nn" hreflang="nn" data-title="Faktorisering" 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/Faktorisering" title="Faktorisering – bokmål (norvegiera)" lang="nb" hreflang="nb" data-title="Faktorisering" 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-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%81%E0%A8%A3%E0%A8%A8%E0%A8%96%E0%A9%B0%E0%A8%A1%E0%A9%80%E0%A8%95%E0%A8%B0%E0%A8%A8" title="ਗੁਣਨਖੰਡੀਕਰਨ – punjabera" lang="pa" hreflang="pa" data-title="ਗੁਣਨਖੰਡੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="punjabera" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_na_czynniki" title="Rozkład na czynniki – poloniera" lang="pl" hreflang="pl" data-title="Rozkład na czynniki" data-language-autonym="Polski" data-language-local-name="poloniera" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Fatora%C3%A7%C3%A3o" title="Fatoração – portugesa" lang="pt" hreflang="pt" data-title="Fatoração" 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/Factorizare" title="Factorizare – errumaniera" lang="ro" hreflang="ro" data-title="Factorizare" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%8F" title="Факторизация – errusiera" lang="ru" hreflang="ru" data-title="Факторизация" data-language-autonym="Русский" data-language-local-name="errusiera" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Factorization" title="Factorization – Simple English" lang="en-simple" hreflang="en-simple" data-title="Factorization" 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/Faktoriz%C3%A1cia" title="Faktorizácia – eslovakiera" lang="sk" hreflang="sk" data-title="Faktorizácia" 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/Faktorizacija" title="Faktorizacija – esloveniera" lang="sl" hreflang="sl" data-title="Faktorizacija" data-language-autonym="Slovenščina" data-language-local-name="esloveniera" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Isirayn" title="Isirayn – somaliera" lang="so" hreflang="so" data-title="Isirayn" data-language-autonym="Soomaaliga" data-language-local-name="somaliera" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D0%B8%D1%98%D0%B0" title="Факторизација – serbiera" lang="sr" hreflang="sr" data-title="Факторизација" 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/Faktorisering" title="Faktorisering – suediera" lang="sv" hreflang="sv" data-title="Faktorisering" 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%95%E0%AE%BE%E0%AE%B0%E0%AE%A3%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AE%9F%E0%AF%81%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AF%81%E0%AE%A4%E0%AE%B2%E0%AF%8D" title="காரணிப்படுத்துதல் – tamilera" lang="ta" hreflang="ta" data-title="காரணிப்படுத்துதல்" data-language-autonym="தமிழ்" data-language-local-name="tamilera" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%A2%E0%B8%81%E0%B8%95%E0%B8%B1%E0%B8%A7%E0%B8%9B%E0%B8%A3%E0%B8%B0%E0%B8%81%E0%B8%AD%E0%B8%9A" title="การแยกตัวประกอบ – thailandiera" lang="th" hreflang="th" data-title="การแยกตัวประกอบ" data-language-autonym="ไทย" data-language-local-name="thailandiera" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/%C3%87arpanlara_ay%C4%B1rma" title="Çarpanlara ayırma – turkiera" lang="tr" hreflang="tr" data-title="Çarpanlara ayırma" 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%A4%D0%B0%D0%BA%D1%82%D0%BE%D1%80%D0%B8%D0%B7%D0%B0%D1%86%D1%96%D1%8F" title="Факторизація – ukrainera" lang="uk" hreflang="uk" data-title="Факторизація" 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/%D8%A7%D8%AC%D8%B2%D8%A7%D8%A6%DB%92_%D8%B6%D8%B1%D8%A8%DB%8C" title="اجزائے ضربی – urdua" lang="ur" hreflang="ur" data-title="اجزائے ضربی" 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/Faktorizatsiya" title="Faktorizatsiya – uzbekera" lang="uz" hreflang="uz" data-title="Faktorizatsiya" 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/Ph%C3%A2n_t%C3%ADch_nh%C3%A2n_t%E1%BB%AD" title="Phân tích nhân tử – vietnamera" lang="vi" hreflang="vi" data-title="Phân tích nhân tử" 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%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3" title="因式分解 – wu txinera" lang="wuu" hreflang="wuu" data-title="因式分解" 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%A7%D7%98%D7%90%D7%A8%D7%99%D7%96%D7%90%D7%A6%D7%99%D7%A2" title="פאקטאריזאציע – yiddisha" lang="yi" hreflang="yi" data-title="פאקטאריזאציע" data-language-autonym="ייִדיש" data-language-local-name="yiddisha" class="interlanguage-link-target"><span>ייִדיש</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3" title="因式分解 – txinera" lang="zh" hreflang="zh" data-title="因式分解" 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%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3" title="因式分解 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="因式分解" 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%9B%A0%E5%BC%8F%E5%88%86%E8%A7%A3" title="因式分解 – kantonera" lang="yue" hreflang="yue" data-title="因式分解" 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/Q188804#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 href="/wiki/Faktorizazio" title="Eduki orrialdea ikusi [c]" accesskey="c"><span>Artikulua</span></a></li><li id="ca-talk" 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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> <a href="/wiki/Matematika" title="Matematika">Matematikan,</a> <b>faktorizazio</b> deritzo adierazpen matematiko bat (<a href="/wiki/Zenbaki" title="Zenbaki">zenbakiak</a>, <a href="/wiki/Polinomio" title="Polinomio">polinomioak</a>, <a href="/wiki/Matrize" title="Matrize">matrizeak</a>...) biderkadura gisa deskonposatzeko teknikari. Hainbat faktorizazio-metodo daude. Helburua da adierazpena sinplifikatzea edo oinarrizko bloketan (<b>faktoretan</b>) berridaztea, adibidez, zenbaki bat <a href="/wiki/Zenbaki_lehen" title="Zenbaki lehen">zenbaki lehenetan</a> (15 zenbakia 3x5 da) edo polinomio bat polinomio laburtezinetan (<i>x</i><sup>2</sup> − 4 polinomioa (<i>x</i> − 2)(<i>x</i> + 2) da) berridaztea. </p><p>Zenbakien faktorizazioaren aurkakoa da horien biderketa eta, polinomio baten faktorizazioarena, aldiz, <a href="/w/index.php?title=Hedapena&action=edit&redlink=1" class="new" title="Hedapena (sortu gabe)">hedapena</a>. Polinomioa faktorizatutakoan sortzen diren faktoreak biderkatuz, polinomio bakar bat lortzen da, terminoen gehiketa dena. Adibidez, 4<i>x</i> <sup>2</sup> termino bat da. </p><p>Zenbaki osoak faktorizatzeko, <a href="/wiki/Aritmetikaren_oinarrizko_teorema" title="Aritmetikaren oinarrizko teorema">aritmetikaren oinarrizko teorema</a> erabiltzen da eta, polinomioen faktorizaziorako, <a href="/wiki/Aljebraren_oinarrizko_teorema" title="Aljebraren oinarrizko teorema">aljebraren oinarrizko teorema</a>. Matrizeak ere faktoriza daitezke matrize berezi batzuen biderkadura gisa. Matrize-faktorizazioaren ohiko adibideek <a href="/wiki/Matrize_ortogonal" title="Matrize ortogonal">matrize ortogonalak</a>, <a href="/w/index.php?title=Matrize_Unitarioa&action=edit&redlink=1" class="new" title="Matrize Unitarioa (sortu gabe)">unitarioak</a> eta <a href="/w/index.php?title=Matrize_Triangular&action=edit&redlink=1" class="new" title="Matrize Triangular (sortu gabe)">triangularrak</a> erabiltzen dituzte. Hainbat mota daude: <a href="/wiki/QR_deskonposizio" class="mw-redirect" title="QR deskonposizio">QR deskonposizioa</a>, <i>LQ</i>, <i>QL</i>, <i>RQ</i> edo <i>RZ</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Zenbaki_osoen_faktorizazioa">Zenbaki osoen faktorizazioa</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=1" title="Aldatu atal hau: «Zenbaki osoen faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=1" title="Edit section's source code: Zenbaki osoen faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Aritmetikaren_oinarrizko_teorema" title="Aritmetikaren oinarrizko teorema">Aritmetikaren oinarrizko teorema</a>ren arabera, bat baino handiagoa den edozein <a href="/wiki/Zenbaki_oso" title="Zenbaki oso">zenbaki oso</a><a href="/wiki/Zenbaki_oso" title="Zenbaki oso">k</a> zenbaki lehenen bidezko faktorizazio bakarra du<a href="/wiki/Zenbaki_osoen_faktorizazio" title="Zenbaki osoen faktorizazio">. Zenbaki osoak faktorizatze</a>ko <a href="/wiki/Algoritmo" title="Algoritmo">algoritmo</a><a href="/wiki/Algoritmo" title="Algoritmo">ak</a> daude, baina oso handiak diren zenbakietarako ez dago algoritmo efiziente klasikorik. </p><p>Zenbakiak faktorizatzeko, hau da, zenbaki bat bere faktore nagusietan deskonposatzeko modu ohikoena probako zatiketa da: faktorizatu nahi den zenbaki osoa zenbaki lehen bakoitzarekin (2, 3, 5, 7, 11, ...) zatitu, eta <a href="/wiki/Zatigarritasun-erregela" title="Zatigarritasun-erregela">zatigarritasun</a><a href="/wiki/Zatigarritasun-erregela" title="Zatigarritasun-erregela">a</a> egiaztatu behar da. Kontuan hartu behar da zenbaki lehen hori ezin dela izan faktorizatu nahi den zenbakiaren <a href="/wiki/Erro_karratu" title="Erro karratu">erro karratu</a><a href="/wiki/Erro_karratu" title="Erro karratu">a</a> baino handiagoa. Hau da, 100 zenbakia faktorizatzeko, 2, 3, 5, 7, 11, …, zenbakiekin zatitu behar dugu baina, <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 {100}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>100</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {100}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe631187125068d1ef7515edad043d10ff3f928a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle {\sqrt {100}}}"></span>=10 denez, 2, 3, 5 eta 7rekin zatitu beharko da soilik. Zatigarritasuna egiaztatzeko, ikusi behar da ia faktorizatu nahi den zenbakia zenbaki lehen batekin zaitzen denean emaitza beste zenbaki oso bat den. 100 zenbakiaren kasuan, 100/2=50 da; beraz, 100 2rekin zatigarria da. 100/3, aldiz, 33.33 da eta, hori zenbaki osoa ez denez, 100 ez da 3rekin zatigarria. Zenbaki lehen batek faktorizatu nahi den zenbakia zatitzen baldin badu, orduan egiaztatu behar da zenbaki horren <a href="/wiki/Berreketa" title="Berreketa">berretura</a> handiagoak faktorizatu beharreko zenbakia zatitzen duen. 100 zenbakiaren kasuan, ikusi da 2 zenbakiak 100 zatitzen duela; beraz, hurrengo urratsa 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 2^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd7711cd907a2d46557a410fb67fc0d84c52ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{2}}"></span>rekin zatigarria den ikustea, eta 100/<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^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd7711cd907a2d46557a410fb67fc0d84c52ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{2}}"></span>=25; beraz, 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 2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}"></span>rekin zatitu behar da: 100/<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^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}"></span>=12.5. Azken hori zenbaki osoa ez denez, 100 ez 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 2^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52e9f8299773e9205d2055998f3c8eb9441877fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{3}}"></span>rekin zatigarria, eta hurrengo zenbaki lehenarekin zatitu beharko da. Alegia, 3, 5 eta 7 zenbakiekin zatitu behar da. </p><p>Zenbait irizpide daude zatigarritasuna aztertzeko: </p> <ul><li>Zatitu nahi den zenbakiaren azken  <a href="/wiki/Zifra" title="Zifra">digitu</a><a href="/wiki/Zifra" title="Zifra">a</a> 2ren <a href="/wiki/Multiplo_(matematika)" title="Multiplo (matematika)">multiplo</a><a href="/wiki/Multiplo_(matematika)" title="Multiplo (matematika)">a</a> bada, orduan, zenbaki osoa 2rekin zatigarria da.</li> <li>Zatitu nahi den zenbakiaren azken  digitua 5en multiploa bada, orduan, zenbaki osoa 5ekin zatigarria da.</li> <li>Zatitu nahi den zenbakiaren digituen batuketa 3ren multiploa bada, zenbaki osoa 3rekin zatigarria da.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Adibidea">Adibidea</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=2" title="Aldatu atal hau: «Adibidea»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=2" title="Edit section's source code: Adibidea"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>n= 1386 zenbakia faktorizatzeko: </p> <ul><li>Hasi probako zatiketa 2 zenbakiarekin. Nabarmena da 1386 <a href="/wiki/Bikoiti" class="mw-redirect" title="Bikoiti">bikoiti</a><a href="/wiki/Bikoiti" class="mw-redirect" title="Bikoiti">a</a> dela; beraz, n = 2 · n', non n' = 1386 / 2 = 693 den. 693 zenbaki <a href="/w/index.php?title=Bakoitia&action=edit&redlink=1" class="new" title="Bakoitia (sortu gabe)">bakoiti</a><a href="/w/index.php?title=Bakoitia&action=edit&redlink=1" class="new" title="Bakoitia (sortu gabe)">a</a> da; beraz, ez dago 2ren berretura handiagorik n zatitzen duenik.</li> <li>Jarraitu probako zatiketa 3 zenbakiarekin, n' = 693 izanik. 6+9+3=18 3ren multiploa da eta, gainera, 3²=9ren multiploa ere bada; beraz, n" = 693 / 9 = 231 / 3 = 77. 7+7=14 ez da 3ren multiploa. Hortaz, joan hurrengo zenbaki lehenera.</li> <li>Hurrengo zenbaki lehena 5 da, eta 77 ez da 5en multiploa.</li> <li>Hurrengo zenbaki lehena 7 da.  n"=77 7rekin zatigarria da, eta n"' = 77/7 = 11.</li> <li>Geratzen den faktorea, n"'=11, lehena da eta, beraz, bukatu da faktorizazioa.</li> <li>1386=2·3²·7·11.</li></ul> <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=Faktorizazio&veaction=edit&section=3" 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=Faktorizazio&action=edit&section=3" title="Edit section's source code: Polinomioen faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Polinomioak faktorizatzeko teknika modernoak azkarrak eta efizienteak dira, baina algoritmo sofistikatuak erabiltzen dituzte (Ikusi <a href="/w/index.php?title=Polinomioen_faktorizazioa&action=edit&redlink=1" class="new" title="Polinomioen faktorizazioa (sortu gabe)">polinomioen faktorizazioa</a>). Teknika horiek ordenagailuek erabiltzen dituzte <a href="/w/index.php?title=Aljebra-sistema&action=edit&redlink=1" class="new" title="Aljebra-sistema (sortu gabe)">aljebra-sistemetan</a>. Eskuzko faktorizazioan, polinomioak maila txikikoak edo mota jakin batekoak izan behar dira. Horregatik, eskuzko teknikak ez dira baliagarriak ordenagailuetan lan egiteko. Artikulu honetan, beraz, eskuzkoak baino ez dira azalduko. </p><p>Adierazpen bat sinpleagoak diren beste batzuen biderketa moduan idaztea da faktorizazioa. “Sinple”  hitzaren esanahia azaldu behar da. Polinomioen faktorizazioan "sinple" hitzak esan nahi du faktoreak hasierakoak baino maila txikiagoko polinomioak izan behar direla. Adibidez, <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 {\displaystyle x^{2}-y=(x+{\sqrt {y}})(x-{\sqrt {y}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{2}-y=(x+{\sqrt {y}})(x-{\sqrt {y}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a24fd8764235ae54147f7a645db22992a74582f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.62ex; height:3.509ex;" alt="{\displaystyle {\displaystyle x^{2}-y=(x+{\sqrt {y}})(x-{\sqrt {y}})}}"></span> bada faktorizazioa, baina ez polinomioen bidezkoa, faktoreak ez baitira polinomioak<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>. Halaber, termino konstante batekin faktorizatzea, <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 {\displaystyle 3x^{2}-6x+12=3(x^{2}-2x+4)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>12</mn> <mo>=</mo> <mn>3</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle 3x^{2}-6x+12=3(x^{2}-2x+4)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ee04d281013bf01f6fc2da6dc1d1455844e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.834ex; height:3.176ex;" alt="{\displaystyle {\displaystyle 3x^{2}-6x+12=3(x^{2}-2x+4)}}"></span> , ez da polinomioen faktorizaziotzat hartzen, faktore batek ez daukalako polinomioak baino maila txikiagoa, berdina baizik<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>. Beste arazo bat faktoreen koefizienteetan dago; izan ere, faktorizazioa egiten dugunean, faktoreen koefizienteak eta polinomioarenak mota berekoak izatea nahi dugu, hau da, zenbaki osoen polinomio bat zenbaki osoko faktoretan deskonposatu nahi dugu, edo <a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">koefiziente erreal</a><a href="/wiki/Zenbaki_erreal" title="Zenbaki erreal">eko</a> polinomioa koefiziente errealetako faktoreetan. Hori ez da beti posible eta, orduan, polinomioa koefiziente horien gainean laburtezina dela esaten da. Adibidez, x<sup>2</sup> – 2 zenbaki osoen gainean laburtezina da, eta x<sup>2</sup> + 4 zenbaki errealen gainean laburtezina. Lehenengo adibidean 1 eta -2 zenbakiak zenbaki erreal modura ikus daitezke; hortaz, <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 {\displaystyle x^{2}-2=(x+{\sqrt {2}})(x-{\sqrt {2}})}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{2}-2=(x+{\sqrt {2}})(x-{\sqrt {2}})}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26fd2d53ae0b4cb82994c270dcce13c608c540f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.641ex; height:3.176ex;" alt="{\displaystyle {\displaystyle x^{2}-2=(x+{\sqrt {2}})(x-{\sqrt {2}})}}"></span>. Horrek adierazten du polinomioa zenbaki errealen gainean faktorizatzen dela. Batzuetan polinomioa zenbaki errealetan <i>zatitzen</i> dela esaten da. Era berean, 1 eta 4 zenbakiak zenbaki konplexu gisa adieraz daitezkeenez, x² + 4 polinomioa zenbaki konplexuen gainean deskonposatzen 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 {\displaystyle x^{2}+4=(x+2i)(x-2i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>4</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{2}+4=(x+2i)(x-2i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70a495067f369227b7c5501815b5d85aa39e9f42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.374ex; height:3.176ex;" alt="{\displaystyle {\displaystyle x^{2}+4=(x+2i)(x-2i)}}"></span>. </p><p><br /> </p><p>Adibidez, 5. mailako P(x) polinomioa 3. mailako baten eta 2. mailako baten biderketa gisa faktoriza daiteke: </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 {\displaystyle P(x)=x^{5}-x^{3}+69x^{2}-20x+16=}{\displaystyle (x^{3}+4x^{2}-x+1)(x^{2}-4x+16)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>69</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>20</mn> <mi>x</mi> <mo>+</mo> <mn>16</mn> <mo>=</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>4</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> <mo stretchy="false">)</mo> <mo stretchy="false">(</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>16</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle P(x)=x^{5}-x^{3}+69x^{2}-20x+16=}{\displaystyle (x^{3}+4x^{2}-x+1)(x^{2}-4x+16)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75e8d7d6e5fc3959450e45a03dc276d51c1caf94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.698ex; height:3.176ex;" alt="{\displaystyle {\displaystyle P(x)=x^{5}-x^{3}+69x^{2}-20x+16=}{\displaystyle (x^{3}+4x^{2}-x+1)(x^{2}-4x+16)}}"></span>. </p><p><a href="/wiki/Aljebraren_oinarrizko_teorema" title="Aljebraren oinarrizko teorema">Aljebraren oinarrizko teorema</a> modu honetan adieraz daiteke: n mailako edozein <a href="/wiki/Polinomio" title="Polinomio">polinomio</a>, koefiziente konplexuak baditu, erabat banatzen da n faktore linealetan. Faktore horien terminoak polinomioaren erroak dira, eta errealak edo konplexuak izan daitezke. Polinomio erreal baten erro konplexu bakoitza bere zenbaki konplexu konjugatuarekin agertzen denez, polinomio erreal guztiak koefiziente errealeko faktore koadratiko lineal eta/edo laburtezinetan banatzen dira. <a href="/wiki/Zenbaki_konplexu" title="Zenbaki konplexu">Zenbaki konplexu</a> <a href="/wiki/Konjugatu_(matematika)" title="Konjugatu (matematika)">konjugatu</a><a href="/wiki/Konjugatu_(matematika)" title="Konjugatu (matematika)">ko</a> bi faktore biderkatzean, koefiziente errealetako <a href="/w/index.php?title=Faktore_koadratiko&action=edit&redlink=1" class="new" title="Faktore koadratiko (sortu gabe)">faktore koadratiko</a> bat lortzen da. </p> <div class="mw-heading mw-heading3"><h3 id="Faktorizazioaren_historia">Faktorizazioaren historia</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=4" title="Aldatu atal hau: «Faktorizazioaren historia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=4" title="Edit section's source code: Faktorizazioaren historia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Ekuazio_koadratiko" class="mw-redirect" title="Ekuazio koadratiko">Ekuazio koadratikoak</a> ebazteko polinomioen faktorizazio-metodoa erabiltzea gauza berria da. Vera Sanford-ek bere <i>A Short History of Mathematics</i> (1930)<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> lanean dioenez, metodo hau 1631an erabili zuen lehenengoz <a href="/wiki/Thomas_Harriot" title="Thomas Harriot">Thomas Harriot</a>-ek.  Nolanahi ere, Harriotek ez zituen kontuan hartu erro karratu negatiboak egon zitezkeela. Harriot 1621ean hil zen eta <i>Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas</i> liburua, haren beste liburu guztiak bezala, haren heriotzaren ostean argitaratu zen. Harrioten metodoa gaur egungoa ez bezalakoa da. Hasteko, Harriotek taulak marrazten zituen <a href="/wiki/Monomio" title="Monomio">monomioen</a>, <a href="/wiki/Binomio" title="Binomio">binomioen</a> eta <a href="/wiki/Trinomio" title="Trinomio">trinomioen</a> gehiketak, kenketak, biderketak eta zatiketak argitzeko. Ondoren, bigarren atalean faktorizazio-metodoaren oinarria ematen duen biderketa bat idazten zuen Harriotek. Berak honako ekuazio hau ezartzen zuen: <i>aa</i> − <i>ba</i> + <i>ca</i> = + <i>bc</i>, eta horrek aurreko biderketarekin bat etorri behar du: </p> <table class="wikitable"> <tbody><tr> <td><i>a</i> − <i>b</i> </td> <td> </td> <td><i>aa</i> − <i>ba</i> </td> <td> </td></tr> <tr> <td> </td> <td>(===)   </td> <td> </td> <td>  (Harriotek  Robert Recorde-ren berdintza luzea erabiltzen      du.) </td></tr> <tr> <td><i>a</i> + <i>c</i> </td> <td> </td> <td><i>ca</i> − <i>bc</i> </td> <td> </td></tr></tbody></table> <p>Horrela,  <i>aa</i> − <i>ba</i> + <i>ca</i> − <i>bc</i> ekuazioaren terminoak faktorizatzen zituen. </p> <div class="mw-heading mw-heading3"><h3 id="Metodo_orokorrak">Metodo orokorrak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=5" title="Aldatu atal hau: «Metodo orokorrak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=5" title="Edit section's source code: Metodo orokorrak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Edozein polinomioren erabateko faktorizazioa egiten duten algoritmoak existitzen dira eta <a href="/w/index.php?title=Sistema_konputazional&action=edit&redlink=1" class="new" title="Sistema konputazional (sortu gabe)">sistema konputazional</a> gehienetan daude. Oso propietate konplexuak dituzte eskuz garatu ahal izateko. Eskuzko kalkuluetarako badaude metodoak, baina askotan ez dira gai laugarren maila baino gehiagoko polinomioen erabateko faktorizazioa lortzeko. </p> <div class="mw-heading mw-heading4"><h4 id="Faktore_komuna">Faktore komuna</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=6" title="Aldatu atal hau: «Faktore komuna»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=6" title="Edit section's source code: Faktore komuna"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Faktorizatzeko teknikarik erabilena “faktore komuna” da eta honetan datza: polinomioaren <a href="/wiki/Zatitzaile_komun_handien" class="mw-redirect" title="Zatitzaile komun handien">zatitzaile komunetako handiena</a> den monomioa aurkitu eta faktore komuna atera. Adibidez<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>: </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 {\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=}{\displaystyle (2x^{3}y^{2})(3)+(2x^{3}y^{2})(4xy)+(2x^{3}y^{2})(-5x^{2}y)=}{\displaystyle (2x^{3}y^{2})(3+4xy-5x^{2}y).}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>8</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mi>y</mi> <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> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mn>4</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mn>5</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=}{\displaystyle (2x^{3}y^{2})(3)+(2x^{3}y^{2})(4xy)+(2x^{3}y^{2})(-5x^{2}y)=}{\displaystyle (2x^{3}y^{2})(3+4xy-5x^{2}y).}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75128062259664b50039afb406134424986be7c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:99.392ex; height:3.176ex;" alt="{\displaystyle {\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=}{\displaystyle (2x^{3}y^{2})(3)+(2x^{3}y^{2})(4xy)+(2x^{3}y^{2})(-5x^{2}y)=}{\displaystyle (2x^{3}y^{2})(3+4xy-5x^{2}y).}}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Multzokatze_bidezko_faktore_komuna">Multzokatze bidezko faktore komuna</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=7" title="Aldatu atal hau: «Multzokatze bidezko faktore komuna»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=7" title="Edit section's source code: Multzokatze bidezko faktore komuna"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Metodo bat erabilgarria dena baina ez duena bermatzen beti funtzionatzen duela, multzokatze bidezko faktore komunare metodoa da. </p><p>Faktorizazio mota honek polinomioaren terminoak bi talde edo gehiagotan kokatzen ditu, horietako bakoitza metodo ezagun baten bidez faktoriza daitekelarik. Faktorizazio horien guztien emaitzak elkartu daitezke jatorrizko adierazpenaren faktorizazioa lortzeko.   </p><p>Adibidez, polinomio hau faktorizatzeko <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 {\displaystyle 4x^{2}+20x+3xy+15y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>20</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>15</mn> <mi>y</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle 4x^{2}+20x+3xy+15y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4cce06d9c40390ff909177221960fc1e521f2f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.85ex; height:3.009ex;" alt="{\displaystyle {\displaystyle 4x^{2}+20x+3xy+15y}}"></span>: </p> <ol><li>Pareko terminoak multzokatu: <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 {\displaystyle (4x^{2}+20x)+(3xy+15y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20</mn> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>15</mn> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle (4x^{2}+20x)+(3xy+15y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4dbeeca1390c7fd33215605a75cc71cc1ea5625" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.469ex; height:3.176ex;" alt="{\displaystyle {\displaystyle (4x^{2}+20x)+(3xy+15y)}}"></span></li> <li>Multzo bakoitza <a href="/wiki/Zatitzaile_komunetako_handien" class="mw-redirect" title="Zatitzaile komunetako handien">zatitzaile komunetako handiena</a><a href="/wiki/Zatitzaile_komunetako_handien" class="mw-redirect" title="Zatitzaile komunetako handien">ren</a> bidez faktorizatu: <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 {\displaystyle 4x(x+5)+3y(x+5)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>x</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle 4x(x+5)+3y(x+5)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e220690eb1bacd5bbec9d07d8714f0194eecdb2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.934ex; height:2.843ex;" alt="{\displaystyle {\displaystyle 4x(x+5)+3y(x+5)}}"></span></li> <li>Binomioaren faktore komuna faktorizatu: <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 {\displaystyle (x+5)(4x+3y)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle (x+5)(4x+3y)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39f371b85dd26da44eda128b8cc821d957ab09b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.602ex; height:2.843ex;" alt="{\displaystyle {\displaystyle (x+5)(4x+3y)}}"></span>.</li></ol> <p>Nahiz eta multzokatzeak ez duen erabateko faktorizazioa erakusten, lau termino izan ditzake, bi binomioen biderkadura direnak (<a href="/w/index.php?title=Arau_banakorra&action=edit&redlink=1" class="new" title="Arau banakorra (sortu gabe)">arau banakorra</a><a href="/w/index.php?title=Arau_banakorra&action=edit&redlink=1" class="new" title="Arau banakorra (sortu gabe)">ren</a> arabera). Hori gertatzekotan, taldekatzeak bai erabateko faktorizazioa izango da. </p> <div class="mw-heading mw-heading4"><h4 id="Faktorearen_teorema">Faktorearen teorema</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=8" title="Aldatu atal hau: «Faktorearen teorema»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=8" title="Edit section's source code: Faktorearen teorema"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Artikulo nagusia</i>: <a href="/w/index.php?title=Faktorearen_teorema&action=edit&redlink=1" class="new" title="Faktorearen teorema (sortu gabe)">Faktorearen teorema</a> </p><p>Aldagai bakarreko polinomio batentzat, <i>p(x)</i>, faktorearen teoremak zera dio: a polinomioaren <a href="/wiki/Erro" class="mw-redirect mw-disambig" title="Erro">erro</a> bat da ( <i>p(a)=0</i>, polinomioaren zeroa deritzona) baldin eta soilik baldin <i>(x-a)</i> p(x)-ren faktorea bada. <i>p(x)</i>-ren faktorizazioaren beste faktorea <a href="/w/index.php?title=Zatiketa_polinomiko&action=edit&redlink=1" class="new" title="Zatiketa polinomiko (sortu gabe)">Zatiketa polinomikoaren</a> edo <a href="/w/index.php?title=Zatiketa_sintetiko&action=edit&redlink=1" class="new" title="Zatiketa sintetiko (sortu gabe)">zatiketa sintetikoaren</a> bidez lortu daiteke. </p><p>Adibidez, polinomio hau 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 {\displaystyle x^{3}-3x+2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{3}-3x+2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c1f0690aee2eff2583fcd3c0362829e4347b86e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.719ex; height:2.843ex;" alt="{\displaystyle {\displaystyle x^{3}-3x+2}}"></span> </p><p>Aztertuz, ikusten da 1 polinomio honen erroa dela (koefizienteen gehiketa 0 da). Orduan, <i>(x-1)</i> polinomioaren faktore bat da. Zatiketaren bidez, honakoa geratzen 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 {\displaystyle x^{3}-3x+2=(x-1)(x^{2}+x-2)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</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>2</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{3}-3x+2=(x-1)(x^{2}+x-2)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fece58af8ca9489aa20b31b18cee275147bd099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.326ex; height:3.176ex;" alt="{\displaystyle {\displaystyle x^{3}-3x+2=(x-1)(x^{2}+x-2)}}"></span>. </p> <div class="mw-heading mw-heading4"><h4 id="Aldagai_baten_kasua,_erroen_propietateak_erabiliz"><span id="Aldagai_baten_kasua.2C_erroen_propietateak_erabiliz"></span>Aldagai baten kasua, erroen propietateak erabiliz</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=9" title="Aldatu atal hau: «Aldagai baten kasua, erroen propietateak erabiliz»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=9" title="Edit section's source code: Aldagai baten kasua, erroen propietateak erabiliz"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aldagai bateko polinomio bat faktore linealetan (lehenengo mailakoak) erabat faktorizatuta badago, erro guztiak ikusgai dira, eta horiek guztiak berriz ere biderkatuz, koefiziente eta erroen arteko erlazioa ikus daiteke. Formalki, erlazio hauei <a href="/w/index.php?title=Vi%C3%A8te-ren_formulak&action=edit&redlink=1" class="new" title="Viète-ren formulak (sortu gabe)">Viète-ren formulak</a> deritze. Formula hauek ez dute polinomioa faktorizatzen baina erroak zeintzuk izan daitezkeen susmoa izaten laguntzen dute. Hala ere, erroei buruzko informazio gehigarria ezagutzen bada, formulekin konbinatu daiteke eta horrela, erroak lortu; beraz, faktorizazioa. </p><p>Adibidez,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> <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 {\displaystyle x^{3}-5x^{2}-16x+80}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>16</mn> <mi>x</mi> <mo>+</mo> <mn>80</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{3}-5x^{2}-16x+80}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb8776036570c467f43edc42730ebeb7a25f2b1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:20.431ex; height:2.843ex;" alt="{\displaystyle {\displaystyle x^{3}-5x^{2}-16x+80}}"></span> faktoriza daiteke jakinda bere erroen batura zero dela. Hartu <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 {\displaystyle r_{1},r_{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle r_{1},r_{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a78065f404db915c65a20c15f3ec13940f71a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.24ex; height:2.009ex;" alt="{\displaystyle {\displaystyle r_{1},r_{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 {\displaystyle r_{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle r_{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb03c6b1b002c5db1118dd0f3d9fede77f839d18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle {\displaystyle r_{3}}}"></span> polinomioaren hiru erroak. Vièteren formulak hauek dira: </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 {\displaystyle {\begin{aligned}r_{1}+r_{2}+r_{3}&=5\\r_{1}r_{2}+r_{2}r_{3}+r_{3}r_{1}&=-16\\r_{1}r_{2}r_{3}&=-80.\end{aligned}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>16</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>80.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle {\begin{aligned}r_{1}+r_{2}+r_{3}&=5\\r_{1}r_{2}+r_{2}r_{3}+r_{3}r_{1}&=-16\\r_{1}r_{2}r_{3}&=-80.\end{aligned}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04b42b2bd6de243a5338ecf78e0c584fddfd2d52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:26.928ex; height:8.843ex;" alt="{\displaystyle {\displaystyle {\begin{aligned}r_{1}+r_{2}+r_{3}&=5\\r_{1}r_{2}+r_{2}r_{3}+r_{3}r_{1}&=-16\\r_{1}r_{2}r_{3}&=-80.\end{aligned}}}}"></span> </p><p>Ikusten 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 {\displaystyle r_{2}+r_{3}=0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle r_{2}+r_{3}=0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e31cbd1ed794bc0e14c545fed56910c79881afa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.307ex; height:2.509ex;" alt="{\displaystyle {\displaystyle r_{2}+r_{3}=0}}"></span> hartuta, <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 {\displaystyle r_{1}=5}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle r_{1}=5}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/386007d4bc9b3569078d5ede049c62aa358a2223" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.364ex; height:2.509ex;" alt="{\displaystyle {\displaystyle r_{1}=5}}"></span> lortzen dela, beraz, beste bi ekuazioak honetara laburtzen ditu: <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 {\displaystyle r_{2}^{2}=16.}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>16.</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle r_{2}^{2}=16.}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a527194031d6ca3c5da36d8cd67f88368ab1da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.173ex; height:3.176ex;" alt="{\displaystyle {\displaystyle r_{2}^{2}=16.}}"></span> </p><p>Modu honetan erroak 5, 4 eta -4 dira eta lortzen 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 {\displaystyle x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>16</mn> <mi>x</mi> <mo>+</mo> <mn>80</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70bb6fc2b4f9ea7428c8cd9985f9305ad2410086" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.602ex; height:3.176ex;" alt="{\displaystyle {\displaystyle x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Patroi_ezagunak">Patroi ezagunak</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=10" title="Aldatu atal hau: «Patroi ezagunak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=10" title="Edit section's source code: Patroi ezagunak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Bi karraturen kenketa: </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 a^{2}-b^{2}=(a+b)(a-b)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d52e4cff7a7157a34586d9a29df412a7d37f574" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.029ex; height:3.176ex;" alt="{\displaystyle a^{2}-b^{2}=(a+b)(a-b)}"></span>. </p><p>Oinarrizko formula hori itxura konplexuagoak dituzten ekuazioetan erabili daiteke, 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 a^{2}+2ab+b^{2}-x^{2}+2xy-y^{2}=(a^{2}+2ab+b^{2})-(x^{2}-2xy+y^{2})=(a+b)^{2}-(x-y)^{2}=(a+b+x-y)(a+b-x+y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>−</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <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> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>−</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+2ab+b^{2}-x^{2}+2xy-y^{2}=(a^{2}+2ab+b^{2})-(x^{2}-2xy+y^{2})=(a+b)^{2}-(x-y)^{2}=(a+b+x-y)(a+b-x+y).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db0e97d1cf28cb8a2fa86d100c23158dec5bcd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:122.955ex; height:3.176ex;" alt="{\displaystyle a^{2}+2ab+b^{2}-x^{2}+2xy-y^{2}=(a^{2}+2ab+b^{2})-(x^{2}-2xy+y^{2})=(a+b)^{2}-(x-y)^{2}=(a+b+x-y)(a+b-x+y).}"></span> </p> <ul><li>Kuboen kenketa edo gehiketa modu honetan faktoriza daiteke:</li></ul> <ol><li>Gehiketa: <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^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1920bead986d253c0a3691cc3b4afd954804af47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.205ex; height:3.176ex;" alt="{\displaystyle a^{3}+b^{3}=(a+b)(a^{2}-ab+b^{2})}"></span>.</li> <li>Kenketa: <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^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2685938142d9c6395fe9b55ae9d5d107674f3ae5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.205ex; height:3.176ex;" alt="{\displaystyle a^{3}-b^{3}=(a-b)(a^{2}+ab+b^{2})}"></span>.</li></ol> <ul><li>n. berreturen kenketa edo gehiketa modu honetan faktoriza daiteke:</li></ul> <p>Izan bedi n edozein zenbaki oso positiboa, kenketaren faktorizazio orokorra hau 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 a^{n}-b^{n}=(a-b)(a^{n-1}+ba^{n-2}+b^{2}a^{n-3}+\ldots +b^{n-2}a+b^{n-1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>a</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}-b^{n}=(a-b)(a^{n-1}+ba^{n-2}+b^{2}a^{n-3}+\ldots +b^{n-2}a+b^{n-1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92513c646815514c6cc9c3a55f8b57ae60238bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.42ex; height:3.176ex;" alt="{\displaystyle a^{n}-b^{n}=(a-b)(a^{n-1}+ba^{n-2}+b^{2}a^{n-3}+\ldots +b^{n-2}a+b^{n-1}).}"></span> </p><p>Gehiketarako bi kasu bereiz daiteke, n bakoitia edo n bikoitia. </p> <ol><li>n bakoitia baldin bada, <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}+b^{n}=(a+b)(a^{n-1}-ba^{n-2}+b^{2}a^{n-3}-\ldots -b^{n-2}a+b^{n-1}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mo>…</mo> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>a</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}+b^{n}=(a+b)(a^{n-1}-ba^{n-2}+b^{2}a^{n-3}-\ldots -b^{n-2}a+b^{n-1}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53877e0a17f41c12107a1cd98eb25bf0900d8a89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:63.42ex; height:3.176ex;" alt="{\displaystyle a^{n}+b^{n}=(a+b)(a^{n-1}-ba^{n-2}+b^{2}a^{n-3}-\ldots -b^{n-2}a+b^{n-1}).}"></span></li> <li>n bikoitia denenan, beste bi kasu berezi daitezke:</li></ol> <p>n 2ren berretura baldin bada,  <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 {\displaystyle a^{n}+b^{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle a^{n}+b^{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f86a5a5221465b6dd2610c4b245610e7b6ff8a00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.505ex; height:2.509ex;" alt="{\displaystyle {\displaystyle a^{n}+b^{n}}}"></span> ezin da faktorizatu. </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 n=m\cdot 2^{k},\ k>0\ eta\ m>1\ bakoitia\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mi>m</mi> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>,</mo> <mtext> </mtext> <mi>k</mi> <mo>></mo> <mn>0</mn> <mtext> </mtext> <mi>e</mi> <mi>t</mi> <mi>a</mi> <mtext> </mtext> <mi>m</mi> <mo>></mo> <mn>1</mn> <mtext> </mtext> <mi>b</mi> <mi>a</mi> <mi>k</mi> <mi>o</mi> <mi>i</mi> <mi>t</mi> <mi>i</mi> <mi>a</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=m\cdot 2^{k},\ k>0\ eta\ m>1\ bakoitia\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3564e367dcccddfd2400408c49d3826258ca7796" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:37.568ex; height:3.009ex;" alt="{\displaystyle n=m\cdot 2^{k},\ k>0\ eta\ m>1\ bakoitia\ }"></span>bada, </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 a^{n}+b^{n}=(a^{2^{k}}+b^{2^{k}})(a^{n-2^{k}}-a^{n-2\cdot 2^{k}}b^{2^{k}}+a^{n-3\cdot 2^{k}}b^{2\cdot 2^{k}}-\ldots -a^{2^{k}}b^{n-2\cdot 2^{k}}+b^{n-2^{k}})=(a^{2^{k}}+b^{2^{k}})\sum _{i=1}^{m}a^{(m-i)2^{k}}(-b^{2^{k}})^{i-1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>−</mo> <mo>…</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo>⋅</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">)</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{n}+b^{n}=(a^{2^{k}}+b^{2^{k}})(a^{n-2^{k}}-a^{n-2\cdot 2^{k}}b^{2^{k}}+a^{n-3\cdot 2^{k}}b^{2\cdot 2^{k}}-\ldots -a^{2^{k}}b^{n-2\cdot 2^{k}}+b^{n-2^{k}})=(a^{2^{k}}+b^{2^{k}})\sum _{i=1}^{m}a^{(m-i)2^{k}}(-b^{2^{k}})^{i-1}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc088780675d74b58070cc5530804dc93279eb94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:116.423ex; height:6.843ex;" alt="{\displaystyle a^{n}+b^{n}=(a^{2^{k}}+b^{2^{k}})(a^{n-2^{k}}-a^{n-2\cdot 2^{k}}b^{2^{k}}+a^{n-3\cdot 2^{k}}b^{2\cdot 2^{k}}-\ldots -a^{2^{k}}b^{n-2\cdot 2^{k}}+b^{n-2^{k}})=(a^{2^{k}}+b^{2^{k}})\sum _{i=1}^{m}a^{(m-i)2^{k}}(-b^{2^{k}})^{i-1}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Polinomioaren_erroen_formula">Polinomioaren erroen formula</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=11" title="Aldatu atal hau: «Polinomioaren erroen formula»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=11" title="Edit section's source code: Polinomioaren erroen formula"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Aldagai bateko bigarren mailako edozein polinomio (modu honetako 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 {\displaystyle ax^{2}+bx+c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle ax^{2}+bx+c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3bd250b833488efdc1044459da2d962aa29a8f7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.629ex; height:2.843ex;" alt="{\displaystyle {\displaystyle ax^{2}+bx+c}}"></span>) zenbaki konplexuen gorputzean faktoriza daiteke formula honen bidez: </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 {\displaystyle ax^{2}+bx+c=a(x-\alpha )(x-\beta )=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right),}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>c</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>β</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mi>b</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <mi>c</mi> </msqrt> </mrow> </mrow> <mrow> <mn>2</mn> <mi>a</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle ax^{2}+bx+c=a(x-\alpha )(x-\beta )=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right),}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93f694151a7910fced2df9935cf5644da1bb660a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.769ex; height:7.509ex;" alt="{\displaystyle {\displaystyle ax^{2}+bx+c=a(x-\alpha )(x-\beta )=a\left(x-{\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\right)\left(x-{\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}\right),}}"></span> </p><p>Formula honi <a href="/w/index.php?title=Formula_koadratiko&action=edit&redlink=1" class="new" title="Formula koadratiko (sortu gabe)">formula koadratikoa</a> deritzo, 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 {\displaystyle \alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle \alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d0af69b7beedb6c8c1303b4bfee3859c16fbfbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle {\displaystyle \alpha }}"></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 {\displaystyle \beta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle \beta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52ff91d951b0e6940f30355efbaa3e32cf596308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle {\displaystyle \beta }}"></span> polinomioaren bi erroak dira (biak errealak edo konplexuak izan daitezke). </p><p>Formula <a href="/w/index.php?title=Kubiko&action=edit&redlink=1" class="new" title="Kubiko (sortu gabe)">kubikoa</a> eta <a href="/w/index.php?title=Kuartikoa&action=edit&redlink=1" class="new" title="Kuartikoa (sortu gabe)">kuartikoa</a> existitzen dira, hala ere, ez dago formularik maila altuagoko polinomioen erroak lortzeko. Kasu horretan, <a href="/wiki/Ruffiniren_erregela" title="Ruffiniren erregela">Ruffini-ren erregela</a> erabili behar da. </p> <div class="mw-heading mw-heading3"><h3 id="Zenbaki_konplexuen_gaineko_faktorizazioa">Zenbaki konplexuen gaineko faktorizazioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=12" title="Aldatu atal hau: «Zenbaki konplexuen gaineko faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=12" title="Edit section's source code: Zenbaki konplexuen gaineko faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Bi_karratuen_batura">Bi karratuen batura</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=13" title="Aldatu atal hau: «Bi karratuen batura»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=13" title="Edit section's source code: Bi karratuen batura"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>a eta b bi zenbaki erreal badira, haien karratuen batura <a href="/wiki/Zenbaki_konplexu" title="Zenbaki konplexu">zenbaki konplexuen</a> biderketa gisa idatz daiteke. Honek faktorizazioaren formula osatzen du:  <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 {\displaystyle a^{2}+b^{2}=(a+bi)(a-bi).}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mi>i</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle a^{2}+b^{2}=(a+bi)(a-bi).}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8483deab700992716c81b6eba95921ca1de973ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.281ex; height:3.176ex;" alt="{\displaystyle {\displaystyle a^{2}+b^{2}=(a+bi)(a-bi).}}"></span> </p><p>Adibidez, <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 {\displaystyle 4x^{2}+49}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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>49</mn> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle 4x^{2}+49}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65920fea5257f4b660600be4777df13a4bd57c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.712ex; height:2.843ex;" alt="{\displaystyle {\displaystyle 4x^{2}+49}}"></span> modu honetan faktoriza 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 {\displaystyle (2x+7i)(2x-7i)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>7</mn> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle (2x+7i)(2x-7i)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f131502738a90107f67abe6c3871be7b76e3a0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.213ex; height:2.843ex;" alt="{\displaystyle {\displaystyle (2x+7i)(2x-7i)}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Matrizeak">Matrizeak</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=14" title="Aldatu atal hau: «Matrizeak»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=14" title="Edit section's source code: Matrizeak"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Matrize baten faktorizazioa, matrize honen deskonposaketa da, beste bi matrize edo gehiagotan, euren forma kanonikoaren arabera. Matrizeak faktorizatzeko hainbat mota daude; bakoitza problema jakin batean erabiltzen delarik. </p> <div class="mw-heading mw-heading3"><h3 id="LU_Faktorizazioa">LU Faktorizazioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=15" title="Aldatu atal hau: «LU Faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=15" title="Edit section's source code: LU Faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A matrize karratu batean aplikagarria.</li> <li>Faktorizazioa: A=LU , non L matrize azpi-triangeluarra den eta U matrize goi-triangeluarra.</li> <li>Notak: LU faktorizazioak Gauss-en metodoa forma matrizialean adierazten du. Izan ere, PA = LU non P permutazio-matrize bat da non L diagonal nagusiko elementu guztiak 1en berdinak diren. Faktorizazioa egoteko baldintza nahikoa A matrizea alderanzgarria izatea da.</li> <li>Ax = b ekuazio linealen sistemaren ebazpena: lehenik, Ly = b ekuazio-sistema ebazten da, eta, ondoren, Ux = y.</li> <li>Baldintza beharrezko eta nahikoa da A-ko minore nagusi guztiak zero ez izatea.</li> <li>Kalkulu-metodoak: Crouten metodoa, U matrize bat lortzen duena, zeinaren diagonaleko elementu guztiak 1 diren.</li></ul> <div class="mw-heading mw-heading3"><h3 id="'"`UNIQ--postMath-0000002B-QINU`"'_Faktorizazioa"><span id=".7F.27.22.60UNIQ--postMath-0000002B-QINU.60.22.27.7F_Faktorizazioa"></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 LDL^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mi>D</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle LDL^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61a66281d5ca0f02cac966650ad3836ec5608e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.479ex; height:2.676ex;" alt="{\displaystyle LDL^{T}}"></span> Faktorizazioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=16" title="Aldatu atal hau: «'"`UNIQ--postMath-0000002B-QINU`"' Faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=16" title="Edit section's source code: '"`UNIQ--postMath-0000002B-QINU`"' Faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A matrize simetriko bati aplika dakioke.</li> <li>Faktorizazioa:<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 {\displaystyle A=LDL^{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>L</mi> <mi>D</mi> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle A=LDL^{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91d8ec1f95eb6a5004976aac7f8599ef0aaa6a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.321ex; height:2.676ex;" alt="{\displaystyle {\displaystyle A=LDL^{T}}}"></span> non L diagonalean batekoak dituen matrize azpi-triangeluarra den 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 {\displaystyle L^{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle L^{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3537b35fe6d9996413d587bb3552c69263f79041" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.972ex; height:2.676ex;" alt="{\displaystyle {\displaystyle L^{T}}}"></span> bere matrize iraulia. Faktorizazioa bakarra da.</li> <li>Baldintza nahikoa da A-ren minore nagusi guztiak zero ez izatea.</li></ul> <div class="mw-heading mw-heading3"><h3 id="QR_Faktorizazioa">QR Faktorizazioa</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=17" title="Aldatu atal hau: «QR Faktorizazioa»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=17" title="Edit section's source code: QR Faktorizazioa"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Aplikagarria: n x m ordenatako edozein A matrizeetanc.</li> <li>Faktorizazioa: <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 {\displaystyle A=QR}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>R</mi> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\displaystyle A=QR}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f91ba11bba000ed59f00fd822602d1aced5dea14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.444ex; height:2.509ex;" alt="{\displaystyle {\displaystyle A=QR}}"></span> non Q m x m matrize ortogonala den, eta R n x m matrize goi-triangeluarra den.</li> <li>Kalkulu-metodoak: QR faktorizazioa A zutabeei aplikatutako Gram-Schmidten ortogonalizazio-prozesuaren bidez, Householderren transformazioen bidez eta Givensen transformazioen bidez kalkula daiteke.</li> <li>Notak: QR faktorizazioa Ax = b ekuazio linealen sistema "ebazteko" erabil daiteke ekuazio kopurua ezezagun kopuruaren desberdina denean.</li></ul> <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=Faktorizazio&veaction=edit&section=18" 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=Faktorizazio&action=edit&section=18" 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-:0-1"><a href="#cite_ref-:0_1-0">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation" id="CITEREFFite1921">Fite, William Benjamin.  (1921). <i>College Algebra (Revised). </i> Boston D.C. Health & Co., 20 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-1143158322" title="Berezi:BookSources/978-1143158322">978-1143158322</a>.</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"> <i>Even if the 3 is thought of as a constant polynomial so that this could be considered a factorization into polynomials.. </i></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" id="CITEREFSandfor2008">Sandfor, Vera.  (2008). <i>A Short History of Mathematics. </i> Read Books <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/9781409727101" title="Berezi:BookSources/9781409727101">9781409727101</a>.</span>.</span> </li> <li id="cite_note-4"><a href="#cite_ref-4">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation" id="CITEREFFite1921">Fite, William Benjamin.  (1921). <i>College Algebra (Revised). </i> Boston: D.C. Heath & Co., 18 or. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="/wiki/Berezi:BookSources/978-1143158322" title="Berezi:BookSources/978-1143158322">978-1143158322</a>.</span>.</span> </li> <li id="cite_note-5"><a href="#cite_ref-5">↑</a> <span class="reference-text"><span class="citation"></span> <span class="citation" id="CITEREFBurnside1960">Burnside, William Snow.  (1960). <i>The Theory of Equations with an introduction to the theory of binary algebraic forms (Volume one).. </i>, 38 or.</span>.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografia">Bibliografia</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Faktorizazio&veaction=edit&section=19" title="Aldatu atal hau: «Bibliografia»" class="mw-editsection-visualeditor"><span>aldatu</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Faktorizazio&action=edit&section=19" title="Edit section's source code: Bibliografia"><span>aldatu iturburu kodea</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="https://en.wikipedia.org/wiki/William_Burnside" class="extiw" title="en:William Burnside">Burnside, William Snow;</a> Panton, Arthur William (1960) [1912], <i>The Theory of Equations with an introduction to the theory of binary algebraic forms (Volume on</i></li> <li><a href="https://en.wikipedia.org/wiki/Leonard_Eugene_Dickson" class="extiw" title="en:Leonard Eugene Dickson">Dickson, Leonard Eugene</a> (1922), First Course in the Theory of Equations, New York: John Wiley & Sons</li> <li>Fite, William Benjamin (1921), College Algebra (Revised), Boston: D. C. Heath & Co.</li> <li><a href="https://en.wikipedia.org/wiki/Felix_Klein" class="extiw" title="en:Felix Klein">Klein, Felix</a> (1925), Elementary Mathematics from an Advanced Standpoint; Arithmetic, Algebra, Analysis, Dover</li> <li>Selby, Samuel M., CRC Standard Mathematical Tables (18th ed.), The Chemical Rubber Co.</li></ul> <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=Faktorizazio&veaction=edit&section=20" 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=Faktorizazio&action=edit&section=20" 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 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