Factorization - Wikipedia

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class="vector-toc-link" href="#History_of_factorization_of_expressions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>History of factorization of expressions</span> </div> </a> <ul id="toc-History_of_factorization_of_expressions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_methods"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>General methods</span> </div> </a> <ul id="toc-General_methods-sublist" class="vector-toc-list"> <li id="toc-Common_factor" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Common_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Common factor</span> </div> </a> <ul id="toc-Common_factor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Grouping" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Grouping"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Grouping</span> </div> </a> <ul id="toc-Grouping-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adding_and_subtracting_terms" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Adding_and_subtracting_terms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Adding and subtracting terms</span> </div> </a> <ul id="toc-Adding_and_subtracting_terms-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Recognizable_patterns" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Recognizable_patterns"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Recognizable patterns</span> </div> </a> <ul id="toc-Recognizable_patterns-sublist" class="vector-toc-list"> <li id="toc-Roots_of_unity" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Roots_of_unity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Roots of unity</span> </div> </a> <ul id="toc-Roots_of_unity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Polynomials" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Polynomials"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Polynomials</span> </div> </a> <button aria-controls="toc-Polynomials-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>Toggle Polynomials subsection</span> </button> <ul id="toc-Polynomials-sublist" class="vector-toc-list"> <li id="toc-Primitive-part_&_content_factorization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primitive-part_&_content_factorization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Primitive-part & content factorization</span> </div> </a> <ul id="toc-Primitive-part_&_content_factorization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_the_factor_theorem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_the_factor_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Using the factor theorem</span> </div> </a> <ul id="toc-Using_the_factor_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rational_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rational_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Rational roots</span> </div> </a> <ul id="toc-Rational_roots-sublist" class="vector-toc-list"> <li id="toc-Quadratic_ac_method" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Quadratic_ac_method"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Quadratic ac method</span> </div> </a> <ul id="toc-Quadratic_ac_method-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Using_formulas_for_polynomial_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_formulas_for_polynomial_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Using formulas for polynomial roots</span> </div> </a> <ul id="toc-Using_formulas_for_polynomial_roots-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Using_relations_between_roots" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Using_relations_between_roots"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.5</span> <span>Using relations between roots</span> </div> </a> <ul id="toc-Using_relations_between_roots-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Unique_factorization_domains" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Unique_factorization_domains"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Unique factorization domains</span> </div> </a> <ul id="toc-Unique_factorization_domains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideals" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Ideals"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Ideals</span> </div> </a> <ul id="toc-Ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Matrices</span> </div> </a> <ul id="toc-Matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" 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="Toggle the table of contents" > <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">Toggle the table of contents</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">Factorization</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="Go to an article in another language. Available in 62 languages" > <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 languages</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 – Afrikaans" lang="af" hreflang="af" data-title="Faktorisasie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%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 – Arabic" lang="ar" hreflang="ar" data-title="تحليل إلى عوامل" data-language-autonym="العربية" data-language-local-name="Arabic" 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 – Assamese" lang="as" hreflang="as" data-title="উৎপাদক বিশ্লেষণ" data-language-autonym="অসমীয়া" data-language-local-name="Assamese" 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 – Azerbaijani" lang="az" hreflang="az" data-title="Faktorizasiya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</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 – Bangla" lang="bn" hreflang="bn" data-title="উৎপাদকে বিশ্লেষণ" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</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 – Bulgarian" lang="bg" hreflang="bg" data-title="Факторизация" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Factoritzaci%C3%B3" title="Factorització – Catalan" lang="ca" hreflang="ca" data-title="Factorització" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</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 – Chuvash" lang="cv" hreflang="cv" data-title="Факторизаци" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Faktorizace" title="Faktorizace – Czech" lang="cs" hreflang="cs" data-title="Faktorizace" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Ffactorau_cysefin" title="Ffactorau cysefin – Welsh" lang="cy" hreflang="cy" data-title="Ffactorau cysefin" data-language-autonym="Cymraeg" data-language-local-name="Welsh" 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 – Danish" lang="da" hreflang="da" data-title="Faktorisering" data-language-autonym="Dansk" data-language-local-name="Danish" 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 – German" lang="de" hreflang="de" data-title="Faktorisierung" data-language-autonym="Deutsch" data-language-local-name="German" 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 – Greek" lang="el" hreflang="el" data-title="Παραγοντοποίηση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorizaci%C3%B3n" title="Factorización – Spanish" lang="es" hreflang="es" data-title="Factorización" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Faktorado" title="Faktorado – Esperanto" lang="eo" hreflang="eo" data-title="Faktorado" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Faktorizazio" title="Faktorizazio – Basque" lang="eu" hreflang="eu" data-title="Faktorizazio" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%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 – Persian" lang="fa" hreflang="fa" data-title="اتحاد و تجزیه" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fo mw-list-item"><a href="https://fo.wikipedia.org/wiki/Valdur" title="Valdur – Faroese" lang="fo" hreflang="fo" data-title="Valdur" data-language-autonym="Føroyskt" data-language-local-name="Faroese" 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 – French" lang="fr" hreflang="fr" data-title="Factorisation" data-language-autonym="Français" data-language-local-name="French" 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 – Galician" lang="gl" hreflang="gl" data-title="Factorización" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</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 – Korean" lang="ko" hreflang="ko" data-title="인수분해" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</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 – Hindi" lang="hi" hreflang="hi" data-title="गुणनखण्ड" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Faktorigo" title="Faktorigo – Ido" lang="io" hreflang="io" data-title="Faktorigo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Faktorisasi" title="Faktorisasi – Indonesian" lang="id" hreflang="id" data-title="Faktorisasi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</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 – Icelandic" lang="is" hreflang="is" data-title="Þáttun" data-language-autonym="Íslenska" data-language-local-name="Icelandic" 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 – Italian" lang="it" hreflang="it" data-title="Fattorizzazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</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 – Hebrew" lang="he" hreflang="he" data-title="פירוק לגורמים" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</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 – Latin" lang="la" hreflang="la" data-title="In factores resolutio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Faktorizavimas" title="Faktorizavimas – Lithuanian" lang="lt" hreflang="lt" data-title="Faktorizavimas" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</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-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Faktoriz%C3%A1ci%C3%B3" title="Faktorizáció – Hungarian" lang="hu" hreflang="hu" data-title="Faktorizáció" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</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 – Macedonian" lang="mk" hreflang="mk" data-title="Факторизација" data-language-autonym="Македонски" data-language-local-name="Macedonian" 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 – Malay" lang="ms" hreflang="ms" data-title="Pemfaktoran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" 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 – Dutch" lang="nl" hreflang="nl" data-title="Factorisatie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</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 – Japanese" lang="ja" hreflang="ja" data-title="因数分解" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Faktorisering" title="Faktorisering – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Faktorisering" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Faktorisering" title="Faktorisering – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Faktorisering" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Faktorizatsiya" title="Faktorizatsiya – Uzbek" lang="uz" hreflang="uz" data-title="Faktorizatsiya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</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 – Punjabi" lang="pa" hreflang="pa" data-title="ਗੁਣਨਖੰਡੀਕਰਨ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" 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 – Polish" lang="pl" hreflang="pl" data-title="Rozkład na czynniki" data-language-autonym="Polski" data-language-local-name="Polish" 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 – Portuguese" lang="pt" hreflang="pt" data-title="Fatoração" data-language-autonym="Português" data-language-local-name="Portuguese" 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 – Romanian" lang="ro" hreflang="ro" data-title="Factorizare" data-language-autonym="Română" data-language-local-name="Romanian" 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 – Russian" lang="ru" hreflang="ru" data-title="Факторизация" data-language-autonym="Русский" data-language-local-name="Russian" 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 – Slovak" lang="sk" hreflang="sk" data-title="Faktorizácia" data-language-autonym="Slovenčina" data-language-local-name="Slovak" 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 – Slovenian" lang="sl" hreflang="sl" data-title="Faktorizacija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" 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 – Somali" lang="so" hreflang="so" data-title="Isirayn" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</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 – Central Kurdish" lang="ckb" hreflang="ckb" data-title="ھاوئەنجام و شیتەڵکردن" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</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 – Serbian" lang="sr" hreflang="sr" data-title="Факторизација" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Tekij%C3%A4" title="Tekijä – Finnish" lang="fi" hreflang="fi" data-title="Tekijä" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Faktorisering" title="Faktorisering – Swedish" lang="sv" hreflang="sv" data-title="Faktorisering" data-language-autonym="Svenska" data-language-local-name="Swedish" 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 – Tamil" lang="ta" hreflang="ta" data-title="காரணிப்படுத்துதல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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 – Thai" lang="th" hreflang="th" data-title="การแยกตัวประกอบ" data-language-autonym="ไทย" data-language-local-name="Thai" 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 – Turkish" lang="tr" hreflang="tr" data-title="Çarpanlara ayırma" data-language-autonym="Türkçe" data-language-local-name="Turkish" 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 – Ukrainian" lang="uk" hreflang="uk" data-title="Факторизація" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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 – Urdu" lang="ur" hreflang="ur" data-title="اجزائے ضربی" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</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 – Vietnamese" lang="vi" hreflang="vi" data-title="Phân tích nhân tử" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</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-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" lang="wuu" hreflang="wuu" data-title="因式分解" data-language-autonym="吴语" data-language-local-name="Wu" 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 – Yiddish" lang="yi" hreflang="yi" data-title="פאקטאריזאציע" data-language-autonym="ייִדיש" data-language-local-name="Yiddish" class="interlanguage-link-target"><span>ייִדיש</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 – Cantonese" lang="yue" hreflang="yue" data-title="因式分解" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</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 – Chinese" lang="zh" hreflang="zh" data-title="因式分解" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</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="Edit interlanguage 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style="display:none">(Mathematical) decomposition into a product</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Factor_(disambiguation)" class="mw-redirect mw-disambig" title="Factor (disambiguation)">Factor (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Factorisatie.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Factorisatie.svg/220px-Factorisatie.svg.png" decoding="async" width="220" height="91" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Factorisatie.svg/330px-Factorisatie.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Factorisatie.svg/440px-Factorisatie.svg.png 2x" data-file-width="337" data-file-height="139" /></a><figcaption>The polynomial <i>x</i><sup>2</sup> + <i>cx</i> + <i>d</i>, where <i>a + b = c</i> and <i>ab = d</i>, can be factorized into (<i>x + a</i>)(<i>x + b</i>).</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>factorization</b> (or <b>factorisation</b>, see <a href="/wiki/American_and_British_English_spelling_differences#-ise,_-ize_(-isation,_-ization)" title="American and British English spelling differences">English spelling differences</a>) or <b>factoring</b> consists of writing a number or another <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical object</a> as a product of several <i><a href="/wiki/Factor_(arithmetic)" class="mw-redirect" title="Factor (arithmetic)">factors</a></i>, usually smaller or simpler objects of the same kind. For example, <span class="texhtml">3 × 5</span> is an <i><a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a></i> of <span class="texhtml">15</span>, and <span class="texhtml">(<i>x</i> – 2)(<i>x</i> + 2)</span> is a <i><a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a></i> of <span class="texhtml"><i>x</i><sup>2</sup> – 4</span>. </p><p>Factorization is not usually considered meaningful within number systems possessing <a href="/wiki/Division_ring" title="Division ring">division</a>, such as the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, since any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span> can be trivially written as <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 (xy)\times (1/y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (xy)\times (1/y)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b22f760956de5c24d8113fcbd8f45190eaf7e475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.425ex; height:2.843ex;" alt="{\displaystyle (xy)\times (1/y)}"></span> whenever <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"></span> is not zero. However, a meaningful factorization for a <a href="/wiki/Rational_number" title="Rational number">rational number</a> or a <a href="/wiki/Rational_function" title="Rational function">rational function</a> can be obtained by writing it in lowest terms and separately factoring its numerator and denominator. </p><p>Factorization was first considered by <a href="/wiki/Greek_mathematics" title="Greek mathematics">ancient Greek mathematicians</a> in the case of integers. They proved the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, which asserts that every positive integer may be factored into a product of <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, which cannot be further factored into integers greater than 1. Moreover, this factorization is unique <a href="/wiki/Up_to" title="Up to">up to</a> the order of the factors. Although <a href="/wiki/Integer_factorization" title="Integer factorization">integer factorization</a> is a sort of inverse to multiplication, it is much more difficult <a href="/wiki/Integer_factorization" title="Integer factorization">algorithmically</a>, a fact which is exploited in the <a href="/wiki/RSA_cryptosystem" class="mw-redirect" title="RSA cryptosystem">RSA cryptosystem</a> to implement <a href="/wiki/Public-key_cryptography" title="Public-key cryptography">public-key cryptography</a>. </p><p><a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization</a> has also been studied for centuries. In elementary algebra, factoring a polynomial reduces the problem of finding its <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> to finding the roots of the factors. Polynomials with coefficients in the integers or in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> possess the <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization property</a>, a version of the fundamental theorem of arithmetic with prime numbers replaced by <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible polynomials</a>. In particular, a <a href="/wiki/Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">univariate polynomial</a> with <a href="/wiki/Complex_number" title="Complex number">complex</a> coefficients admits a unique (up to ordering) factorization into <a href="/wiki/Linear_polynomial" class="mw-redirect" title="Linear polynomial">linear polynomials</a>: this is a version of the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a>. In this case, the factorization can be done with <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">root-finding algorithms</a>. The case of polynomials with integer coefficients is fundamental for <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a>. There are efficient computer <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for computing (complete) factorizations within the ring of polynomials with rational number coefficients (see <a href="/wiki/Factorization_of_polynomials" title="Factorization of polynomials">factorization of polynomials</a>). </p><p>A <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> possessing the unique factorization property is called a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a>. There are <a href="/wiki/Number_system" class="mw-redirect" title="Number system">number systems</a>, such as certain <a href="/wiki/Ring_of_algebraic_integers" class="mw-redirect" title="Ring of algebraic integers">rings of algebraic integers</a>, which are not unique factorization domains. However, rings of algebraic integers satisfy the weaker property of <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a>: <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> factor uniquely into <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a>. </p><p><i>Factorization</i> may also refer to more general decompositions of a mathematical object into the product of smaller or simpler objects. For example, every function may be factored into the composition of a <a href="/wiki/Surjective_function" title="Surjective function">surjective function</a> with an <a href="/wiki/Injective_function" title="Injective function">injective function</a>. <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a> possess many kinds of <a href="/wiki/Matrix_factorization" class="mw-redirect" title="Matrix factorization">matrix factorizations</a>. For example, every matrix has a unique <a href="/wiki/LU_decomposition" title="LU decomposition">LUP factorization</a> as a product of a <a href="/wiki/Lower_triangular_matrix" class="mw-redirect" title="Lower triangular matrix">lower triangular matrix</a> <span class="texhtml mvar" style="font-style:italic;">L</span> with all diagonal entries equal to one, an <a href="/wiki/Upper_triangular_matrix" class="mw-redirect" title="Upper triangular matrix">upper triangular matrix</a> <span class="texhtml mvar" style="font-style:italic;">U</span>, and a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a> <span class="texhtml mvar" style="font-style:italic;">P</span>; this is a matrix formulation of <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Integers">Integers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=1" title="Edit section: Integers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Integer_factorization" title="Integer factorization">Integer factorization</a></div> <p>By the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>, every <a href="/wiki/Integer" title="Integer">integer</a> greater than 1 has a unique (up to the order of the factors) factorization into <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, which are those integers which cannot be further factorized into the product of integers greater than one. </p><p>For computing the factorization of an integer <span class="texhtml mvar" style="font-style:italic;">n</span>, one needs an <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for finding a <a href="/wiki/Divisor" title="Divisor">divisor</a> <span class="texhtml mvar" style="font-style:italic;">q</span> of <span class="texhtml mvar" style="font-style:italic;">n</span> or deciding that <span class="texhtml mvar" style="font-style:italic;">n</span> is prime. When such a divisor is found, the repeated application of this algorithm to the factors <span class="texhtml mvar" style="font-style:italic;">q</span> and <span class="texhtml"><i>n</i> / <i>q</i></span> gives eventually the complete factorization of <span class="texhtml mvar" style="font-style:italic;">n</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>For finding a divisor <span class="texhtml mvar" style="font-style:italic;">q</span> of <span class="texhtml mvar" style="font-style:italic;">n</span>, if any, it suffices to test all values of <span class="texhtml mvar" style="font-style:italic;">q</span> such that <span class="texhtml">1 < <i>q</i></span> and <span class="texhtml"><i>q</i><sup>2</sup> ≤ <i>n</i></span>. In fact, if <span class="texhtml"><i>r</i></span> is a divisor of <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml"><i>r</i><sup>2</sup> > <i>n</i></span>, then <span class="texhtml"><i>q</i> = <i>n</i> / <i>r</i></span> is a divisor of <span class="texhtml mvar" style="font-style:italic;">n</span> such that <span class="texhtml"><i>q</i><sup>2</sup> ≤ <i>n</i></span>. </p><p>If one tests the values of <span class="texhtml mvar" style="font-style:italic;">q</span> in increasing order, the first divisor that is found is necessarily a prime number, and the <i>cofactor</i> <span class="texhtml"><i>r</i> = <i>n</i> / <i>q</i></span> cannot have any divisor smaller than <span class="texhtml mvar" style="font-style:italic;">q</span>. For getting the complete factorization, it suffices thus to continue the algorithm by searching a divisor of <span class="texhtml mvar" style="font-style:italic;">r</span> that is not smaller than <span class="texhtml mvar" style="font-style:italic;">q</span> and not greater than <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>r</i></span></span></span>. </p><p>There is no need to test all values of <span class="texhtml mvar" style="font-style:italic;">q</span> for applying the method. In principle, it suffices to test only prime divisors. This needs to have a table of prime numbers that may be generated for example with the <a href="/wiki/Sieve_of_Eratosthenes" title="Sieve of Eratosthenes">sieve of Eratosthenes</a>. As the method of factorization does essentially the same work as the sieve of Eratosthenes, it is generally more efficient to test for a divisor only those numbers for which it is not immediately clear whether they are prime or not. Typically, one may proceed by testing 2, 3, 5, and the numbers > 5, whose last digit is 1, 3, 7, 9 and the sum of digits is not a multiple of 3. </p><p>This method works well for factoring small integers, but is inefficient for larger integers. For example, <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Pierre de Fermat</a> was unable to discover that the 6th <a href="/wiki/Fermat_number" title="Fermat number">Fermat number</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+2^{2^{5}}=1+2^{32}=4\,294\,967\,297}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo>+</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msup> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace" /> <mn>294</mn> <mspace width="thinmathspace" /> <mn>967</mn> <mspace width="thinmathspace" /> <mn>297</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+2^{2^{5}}=1+2^{32}=4\,294\,967\,297}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/824582b600570bb4f367d429de89a4cac028e24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:33.075ex; height:3.176ex;" alt="{\displaystyle 1+2^{2^{5}}=1+2^{32}=4\,294\,967\,297}"></span></dd></dl> <p>is not a prime number. In fact, applying the above method would require more than <span class="nowrap"><span data-sort-value="7004100000000000000♠"></span>10<span style="margin-left:.25em;">000</span> divisions</span>, for a number that has 10 <a href="/wiki/Decimal_digit" class="mw-redirect" title="Decimal digit">decimal digits</a>. </p><p>There are more efficient factoring algorithms. However they remain relatively inefficient, as, with the present state of the art, one cannot factorize, even with the more powerful computers, a number of 500 decimal digits that is the product of two randomly chosen prime numbers. This ensures the security of the <a href="/wiki/RSA_cryptosystem" class="mw-redirect" title="RSA cryptosystem">RSA cryptosystem</a>, which is widely used for secure <a href="/wiki/Internet" title="Internet">internet</a> communication. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For factoring <span class="texhtml"><i>n</i> = 1386</span> into primes: </p> <ul><li>Start with division by 2: the number is even, and <span class="texhtml"><i>n</i> = 2 · 693</span>. Continue with 693, and 2 as a first divisor candidate.</li> <li>693 is odd (2 is not a divisor), but is a multiple of 3: one has <span class="texhtml">693 = 3 · 231</span> and <span class="texhtml"><i>n</i> = 2 · 3 · 231</span>. Continue with 231, and 3 as a first divisor candidate.</li> <li>231 is also a multiple of 3: one has <span class="texhtml">231 = 3 · 77</span>, and thus <span class="texhtml"><i>n</i> = 2 · 3<sup>2</sup> · 77</span>. Continue with 77, and 3 as a first divisor candidate.</li> <li>77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has <span class="texhtml">77 = 7 · 11</span>, and thus <span class="texhtml"><i>n</i> = 2 · 3<sup>2</sup> · 7 · 11</span>. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.</li> <li>As <span class="texhtml">7<sup>2</sup> > 11</span>, one has finished. Thus 11 is prime, and the prime factorization is</li></ul> <dl><dd><span class="texhtml">1386 = 2 · 3<sup>2</sup> · 7 · 11</span>.</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Expressions">Expressions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=3" title="Edit section: Expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Manipulating <a href="/wiki/Expression_(mathematics)" title="Expression (mathematics)">expressions</a> is the basis of <a href="/wiki/Algebra" title="Algebra">algebra</a>. Factorization is one of the most important methods for expression manipulation for several reasons. If one can put an <a href="/wiki/Equation" title="Equation">equation</a> in a factored form <span class="texhtml"><i>E</i>⋅<i>F</i> = 0</span>, then the problem of solving the equation splits into two independent (and generally easier) problems <span class="texhtml"><i>E</i> = 0</span> and <span class="texhtml"><i>F</i> = 0</span>. When an expression can be factored, the factors are often much simpler, and may thus offer some insight on the problem. For example, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-ax^{2}-bx^{2}-cx^{2}+abx+acx+bcx-abc}"> <semantics> <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> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>b</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>c</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>a</mi> <mi>b</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>c</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>c</mi> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}-ax^{2}-bx^{2}-cx^{2}+abx+acx+bcx-abc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c64b0d92d8a426afd0c72c7e1bc8099b5a0c6d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:46.344ex; height:2.843ex;" alt="{\displaystyle x^{3}-ax^{2}-bx^{2}-cx^{2}+abx+acx+bcx-abc}"></span></dd></dl> <p>having 16 multiplications, 4 subtractions and 3 additions, may be factored into the much simpler expression </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-a)(x-b)(x-c),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-a)(x-b)(x-c),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f93a827512979f80a7d12a3b4916a25d20e87814" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.819ex; height:2.843ex;" alt="{\displaystyle (x-a)(x-b)(x-c),}"></span></dd></dl> <p>with only two multiplications and three subtractions. Moreover, the factored form immediately gives <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> <i>x</i> = <i>a</i>,<i>b</i>,<i>c</i> as the roots of the polynomial. </p><p>On the other hand, factorization is not always possible, and when it is possible, the factors are not always simpler. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{10}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{10}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e32efe6213739d15b7ca24845f5b08dbb27004c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.209ex; height:2.843ex;" alt="{\displaystyle x^{10}-1}"></span> can be factored into two <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible factors</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1a88d34243b98b57c4df9db5724f61b59a4b9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.333ex; height:2.343ex;" alt="{\displaystyle x-1}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{9}+x^{8}+\cdots +x^{2}+x+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>9</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>8</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{9}+x^{8}+\cdots +x^{2}+x+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd02ceebd9238210855ccc96f631ad6cee184e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:26.569ex; height:2.843ex;" alt="{\displaystyle x^{9}+x^{8}+\cdots +x^{2}+x+1}"></span>. </p><p>Various methods have been developed for finding factorizations; some are described <a href="#General_methods">below</a>. </p><p>Solving <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equations</a> may be viewed as a problem of <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a>. In fact, the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> can be stated as follows: every <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> in <span class="texhtml mvar" style="font-style:italic;">x</span> of degree <span class="texhtml"><i>n</i></span> with <a href="/wiki/Complex_number" title="Complex number">complex</a> coefficients may be factorized into <span class="texhtml"><i>n</i></span> linear factors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-a_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-a_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f23e4e61dc125b94da0df1586c646a0581fe468" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.846ex; height:2.343ex;" alt="{\displaystyle x-a_{i},}"></span> for <span class="texhtml"><i>i</i> = 1, ..., <i>n</i></span>, where the <span class="texhtml"><i>a</i><sub><i>i</i></sub></span>s are the roots of the polynomial.<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> Even though the structure of the factorization is known in these cases, the <span class="texhtml"><i>a</i><sub><i>i</i></sub></span>s generally cannot be computed in terms of radicals (<i>n</i><sup>th</sup> roots), by the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a>. In most cases, the best that can be done is computing <a href="/wiki/Approximation" title="Approximation">approximate values</a> of the roots with a <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">root-finding algorithm</a>. </p> <div class="mw-heading mw-heading3"><h3 id="History_of_factorization_of_expressions">History of factorization of expressions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=4" title="Edit section: History of factorization of expressions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The systematic use of algebraic manipulations for simplifying expressions (more specifically <a href="/wiki/Equation" title="Equation">equations</a>) may be dated to 9th century, with <a href="/wiki/Muhammad_ibn_Musa_al-Khwarizmi" class="mw-redirect" title="Muhammad ibn Musa al-Khwarizmi">al-Khwarizmi</a>'s book <i><a href="/wiki/The_Compendious_Book_on_Calculation_by_Completion_and_Balancing" class="mw-redirect" title="The Compendious Book on Calculation by Completion and Balancing">The Compendious Book on Calculation by Completion and Balancing</a></i>, which is titled with two such types of manipulation. </p><p>However, even for solving <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equations</a>, the factoring method was not used before <a href="/wiki/Thomas_Harriot" title="Thomas Harriot">Harriot</a>'s work published in 1631, ten years after his death.<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> In his book <i>Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas</i>, Harriot drew tables for addition, subtraction, multiplication and division of <a href="/wiki/Monomial" title="Monomial">monomials</a>, <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomials</a>, and <a href="/wiki/Trinomial" title="Trinomial">trinomials</a>. Then, in a second section, he set up the equation <span class="texhtml"><i>aa</i> − <i>ba</i> + <i>ca</i> = + <i>bc</i></span>, and showed that this matches the form of multiplication he had previously provided, giving the factorization <span class="texhtml">(<i>a</i> − <i>b</i>)(<i>a</i> + <i>c</i>)</span>.<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> <div class="mw-heading mw-heading3"><h3 id="General_methods">General methods</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=5" title="Edit section: General methods"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following methods apply to any expression that is a sum, or that may be transformed into a sum. Therefore, they are most often applied to <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, though they also may be applied when the terms of the sum are not <a href="/wiki/Monomial" title="Monomial">monomials</a>, that is, the terms of the sum are a product of variables and constants. </p> <div class="mw-heading mw-heading4"><h4 id="Common_factor">Common factor</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=6" title="Edit section: Common factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It may occur that all terms of a sum are products and that some factors are common to all terms. In this case, the <a href="/wiki/Distributive_property" title="Distributive property">distributive law</a> allows factoring out this common factor. If there are several such common factors, it is preferable to divide out the greatest such common factor. Also, if there are integer coefficients, one may factor out the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of these coefficients. </p><p>For example,<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-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=2x^{3}y^{2}(3+4xy-5x^{2}y),}"> <semantics> <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> <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> <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> <annotation encoding="application/x-tex">{\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=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/02ee55ba927ad700686b73eb255debbd7f281ed7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.635ex; height:3.176ex;" alt="{\displaystyle 6x^{3}y^{2}+8x^{4}y^{3}-10x^{5}y^{3}=2x^{3}y^{2}(3+4xy-5x^{2}y),}"></span> since 2 is the greatest common divisor of 6, 8, and 10, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}y^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{3}y^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54f0111f65642df6a4585d711dca0e6f435e6eca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.599ex; height:3.009ex;" alt="{\displaystyle x^{3}y^{2}}"></span> divides all terms. </p> <div class="mw-heading mw-heading4"><h4 id="Grouping">Grouping</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=7" title="Edit section: Grouping"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Grouping terms may allow using other methods for getting a factorization. </p><p>For example, to factor <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x^{2}+20x+3xy+15y,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>15</mn> <mi>y</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}+20x+3xy+15y,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eedbe6ff472e98b6be9a8e2a9562af034345cd5d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.497ex; height:3.009ex;" alt="{\displaystyle 4x^{2}+20x+3xy+15y,}"></span> one may remark that the first two terms have a common factor <span class="texhtml mvar" style="font-style:italic;">x</span>, and the last two terms have the common factor <span class="texhtml mvar" style="font-style:italic;">y</span>. Thus <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x^{2}+20x+3xy+15y=(4x^{2}+20x)+(3xy+15y)=4x(x+5)+3y(x+5).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>15</mn> <mi>y</mi> <mo>=</mo> <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> <mo>=</mo> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}+20x+3xy+15y=(4x^{2}+20x)+(3xy+15y)=4x(x+5)+3y(x+5).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b6c948f5892f6adab475a44cbc728b421fba44" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:78.097ex; height:3.176ex;" alt="{\displaystyle 4x^{2}+20x+3xy+15y=(4x^{2}+20x)+(3xy+15y)=4x(x+5)+3y(x+5).}"></span> Then a simple inspection shows the common factor <span class="texhtml"><i>x</i> + 5</span>, leading to the factorization <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4x^{2}+20x+3xy+15y=(4x+3y)(x+5).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>20</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mn>15</mn> <mi>y</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4x^{2}+20x+3xy+15y=(4x+3y)(x+5).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad4f4be5185d8a2ca2591e0e11330e9fbe9b4eda" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.197ex; height:3.176ex;" alt="{\displaystyle 4x^{2}+20x+3xy+15y=(4x+3y)(x+5).}"></span> </p><p>In general, this works for sums of 4 terms that have been obtained as the product of two <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomials</a>. Although not frequently, this may work also for more complicated examples. </p> <div class="mw-heading mw-heading4"><h4 id="Adding_and_subtracting_terms">Adding and subtracting terms</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=8" title="Edit section: Adding and subtracting terms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sometimes, some term grouping reveals part of a <a href="#Recognizable_patterns">recognizable pattern</a>. It is then useful to add and subtract terms to complete the pattern. </p><p>A typical use of this is the <a href="/wiki/Completing_the_square" title="Completing the square">completing the square</a> method for getting the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>. </p><p>Another example is the factorization of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449600c47ff9e81b9b1ac284f2093d1ed2cd2496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.034ex; height:2.843ex;" alt="{\displaystyle x^{4}+1.}"></span> If one introduces the non-real <a href="/wiki/Square_root_of_%E2%80%931" class="mw-redirect" title="Square root of –1">square root of –1</a>, commonly denoted <span class="texhtml mvar" style="font-style:italic;">i</span>, then one has a <a href="/wiki/Difference_of_squares" class="mw-redirect" title="Difference of squares">difference of squares</a> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1=(x^{2}+i)(x^{2}-i).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>i</mi> <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>i</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1=(x^{2}+i)(x^{2}-i).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5f6ba4802a992b3f82ef4d35fca84e621c2d4c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.804ex; height:3.176ex;" alt="{\displaystyle x^{4}+1=(x^{2}+i)(x^{2}-i).}"></span> However, one may also want a factorization with <a href="/wiki/Real_number" title="Real number">real number</a> coefficients. By adding and subtracting <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b2f051ba7ee6ca7deda628b79a661f8dc0365b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.193ex; height:3.009ex;" alt="{\displaystyle 2x^{2},}"></span> and grouping three terms together, one may recognize the square of a <a href="/wiki/Binomial_(polynomial)" title="Binomial (polynomial)">binomial</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1=(x^{4}+2x^{2}+1)-2x^{2}=(x^{2}+1)^{2}-\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {2}}+1\right)\left(x^{2}-x{\sqrt {2}}+1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1=(x^{4}+2x^{2}+1)-2x^{2}=(x^{2}+1)^{2}-\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {2}}+1\right)\left(x^{2}-x{\sqrt {2}}+1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bfc49d86213dd741f157d67d9b5c4b2083b111a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:89.346ex; height:3.676ex;" alt="{\displaystyle x^{4}+1=(x^{4}+2x^{2}+1)-2x^{2}=(x^{2}+1)^{2}-\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {2}}+1\right)\left(x^{2}-x{\sqrt {2}}+1\right).}"></span> Subtracting and adding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bee032eda968a3b2b16bdd6d98f6b6a613de07b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.546ex; height:2.676ex;" alt="{\displaystyle 2x^{2}}"></span> also yields the factorization: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1=(x^{4}-2x^{2}+1)+2x^{2}=(x^{2}-1)^{2}+\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {-2}}-1\right)\left(x^{2}-x{\sqrt {-2}}-1\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>2</mn> </msqrt> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1=(x^{4}-2x^{2}+1)+2x^{2}=(x^{2}-1)^{2}+\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {-2}}-1\right)\left(x^{2}-x{\sqrt {-2}}-1\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/740a25bac2e858f26fd79140a218487ba39ba837" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:92.962ex; height:3.676ex;" alt="{\displaystyle x^{4}+1=(x^{4}-2x^{2}+1)+2x^{2}=(x^{2}-1)^{2}+\left(x{\sqrt {2}}\right)^{2}=\left(x^{2}+x{\sqrt {-2}}-1\right)\left(x^{2}-x{\sqrt {-2}}-1\right).}"></span> These factorizations work not only over the complex numbers, but also over any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, where either –1, 2 or –2 is a square. In a <a href="/wiki/Finite_field" title="Finite field">finite field</a>, the product of two non-squares is a square; this implies that the <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c6686458d523659ceffb9ccb73d1dbe39c8ff98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.034ex; height:3.009ex;" alt="{\displaystyle x^{4}+1,}"></span> which is <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> over the integers, is reducible <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modulo</a> every <a href="/wiki/Prime_number" title="Prime number">prime number</a>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1\equiv (x+1)^{4}{\pmod {2}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>≡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1\equiv (x+1)^{4}{\pmod {2}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeaac2f28a1f43da149186375aa6a8ca8ff76d39" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.175ex; height:3.176ex;" alt="{\displaystyle x^{4}+1\equiv (x+1)^{4}{\pmod {2}};}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1\equiv (x^{2}+x-1)(x^{2}-x-1){\pmod {3}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>≡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1\equiv (x^{2}+x-1)(x^{2}-x-1){\pmod {3}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37ecb918c35477ef18bc9c90a046a023897e377d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.711ex; height:3.176ex;" alt="{\displaystyle x^{4}+1\equiv (x^{2}+x-1)(x^{2}-x-1){\pmod {3}},}"></span>since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1^{2}\equiv -2{\pmod {3}};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mo>−</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1^{2}\equiv -2{\pmod {3}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1a9593dedd14a72146620fd0b83a0a17c4b4128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.779ex; height:3.176ex;" alt="{\displaystyle 1^{2}\equiv -2{\pmod {3}};}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1\equiv (x^{2}+2)(x^{2}-2){\pmod {5}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <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> <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>2</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1\equiv (x^{2}+2)(x^{2}-2){\pmod {5}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00203053f064a60075a46c64cdcf2c3a3e6d4f73" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.371ex; height:3.176ex;" alt="{\displaystyle x^{4}+1\equiv (x^{2}+2)(x^{2}-2){\pmod {5}},}"></span>since <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}\equiv -1{\pmod {5}};}"> <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> <mo>≡</mo> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>5</mn> <mo stretchy="false">)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}\equiv -1{\pmod {5}};}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a07f6e44eb461b97135a22df53ce8fc6821bd00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.779ex; height:3.176ex;" alt="{\displaystyle 2^{2}\equiv -1{\pmod {5}};}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{4}+1\equiv (x^{2}+3x+1)(x^{2}-3x+1){\pmod {7}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> <mo>≡</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <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>3</mn> <mi>x</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{4}+1\equiv (x^{2}+3x+1)(x^{2}-3x+1){\pmod {7}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00c2f73e2c493a2e2bef2ac4b6a51f71b52360d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.036ex; height:3.176ex;" alt="{\displaystyle x^{4}+1\equiv (x^{2}+3x+1)(x^{2}-3x+1){\pmod {7}},}"></span>since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{2}\equiv 2{\pmod {7}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≡</mo> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mspace width="1em" /> <mo stretchy="false">(</mo> <mi>mod</mi> <mspace width="0.333em" /> <mn>7</mn> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3^{2}\equiv 2{\pmod {7}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b71c5b33120af636dfbdcc6346fadb94090af521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.971ex; height:3.176ex;" alt="{\displaystyle 3^{2}\equiv 2{\pmod {7}}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Recognizable_patterns">Recognizable patterns</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=9" title="Edit section: Recognizable patterns"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many <a href="/wiki/Identity_(mathematics)" title="Identity (mathematics)">identities</a> provide an equality between a sum and a product. The above methods may be used for letting the sum side of some identity appear in an expression, which may therefore be replaced by a product. </p><p>Below are identities whose left-hand sides are commonly used as patterns (this means that the variables <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span> that appear in these identities may represent any subexpression of the expression that has to be factorized).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Difference_of_squares_and_cubes_visual_proof.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Difference_of_squares_and_cubes_visual_proof.svg/220px-Difference_of_squares_and_cubes_visual_proof.svg.png" decoding="async" width="220" height="293" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/03/Difference_of_squares_and_cubes_visual_proof.svg/330px-Difference_of_squares_and_cubes_visual_proof.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/03/Difference_of_squares_and_cubes_visual_proof.svg/440px-Difference_of_squares_and_cubes_visual_proof.svg.png 2x" data-file-width="512" data-file-height="683" /></a><figcaption>Visual proof of the differences between two squares and two cubes</figcaption></figure> <ul><li><dl><dt><a href="/wiki/Difference_of_two_squares" title="Difference of two squares">Difference of two squares</a></dt></dl></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2}-F^{2}=(E+F)(E-F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2}-F^{2}=(E+F)(E-F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0509844d59d7b3de8c0837c74703be5ff1b6699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.988ex; height:3.176ex;" alt="{\displaystyle E^{2}-F^{2}=(E+F)(E-F)}"></span></dd></dl></dd> <dd>For example, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}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).\end{aligned}}}"> <semantics> <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> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> </mtd> <mtd> <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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}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).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e819f1e465a7ebd1b2dfa6546cfd362e3bd6615" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:41.735ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}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).\end{aligned}}}"></span></dd></dl></dd></dl> <ul><li><dl><dt>Sum/difference of two cubes</dt></dl></li></ul> <p><span class="anchor" id="Sum/difference_of_two_cubes"></span> </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{3}+F^{3}=(E+F)(E^{2}-EF+F^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>E</mi> <mi>F</mi> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{3}+F^{3}=(E+F)(E^{2}-EF+F^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155064b67f3dc28a13c3e5bd91b935ec2b134ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.546ex; height:3.176ex;" alt="{\displaystyle E^{3}+F^{3}=(E+F)(E^{2}-EF+F^{2})}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{3}-F^{3}=(E-F)(E^{2}+EF+F^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>E</mi> <mi>F</mi> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{3}-F^{3}=(E-F)(E^{2}+EF+F^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/feae1f3a95416014d769cf1e0f0d0888f6674697" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.546ex; height:3.176ex;" alt="{\displaystyle E^{3}-F^{3}=(E-F)(E^{2}+EF+F^{2})}"></span></dd></dl></dd></dl> <ul><li><dl><dt>Difference of two fourth powers</dt></dl></li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E^{4}-F^{4}&=(E^{2}+F^{2})(E^{2}-F^{2})\\&=(E^{2}+F^{2})(E+F)(E-F)\end{aligned}}}"> <semantics> <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> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E^{4}-F^{4}&=(E^{2}+F^{2})(E^{2}-F^{2})\\&=(E^{2}+F^{2})(E+F)(E-F)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c53c957143a17e2888c5e0f0a25efd3fe23b446a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:39.106ex; height:6.509ex;" alt="{\displaystyle {\begin{aligned}E^{4}-F^{4}&=(E^{2}+F^{2})(E^{2}-F^{2})\\&=(E^{2}+F^{2})(E+F)(E-F)\end{aligned}}}"></span></dd></dl></dd></dl> <ul><li><dl><dt>Sum/difference of two <span class="texhtml mvar" style="font-style:italic;">n</span>th powers</dt></dl></li></ul> <dl><dd>In the following identities, the factors may often be further factorized: <ul><li><dl><dt>Difference, even exponent</dt></dl></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{2n}-F^{2n}=(E^{n}+F^{n})(E^{n}-F^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{2n}-F^{2n}=(E^{n}+F^{n})(E^{n}-F^{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a563d0f3e95fe5383a8480515a8476106b70c630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.019ex; height:3.176ex;" alt="{\displaystyle E^{2n}-F^{2n}=(E^{n}+F^{n})(E^{n}-F^{n})}"></span></dd></dl> <ul><li><dl><dt>Difference, even or odd exponent</dt></dl></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}-F^{n}=(E-F)(E^{n-1}+E^{n-2}F+E^{n-3}F^{2}+\cdots +EF^{n-2}+F^{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo>+</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>E</mi> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}-F^{n}=(E-F)(E^{n-1}+E^{n-2}F+E^{n-3}F^{2}+\cdots +EF^{n-2}+F^{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8ee0a13b863a16fce7bd23d753b10ef2aca223b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.876ex; height:3.176ex;" alt="{\displaystyle E^{n}-F^{n}=(E-F)(E^{n-1}+E^{n-2}F+E^{n-3}F^{2}+\cdots +EF^{n-2}+F^{n-1})}"></span></dd> <dd>This is an example showing that the factors may be much larger than the sum that is factorized.</dd></dl> <ul><li><dl><dt>Sum, odd exponent</dt></dl></li></ul> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}+F^{n}=(E+F)(E^{n-1}-E^{n-2}F+E^{n-3}F^{2}-\cdots -EF^{n-2}+F^{n-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mi>F</mi> <mo>+</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>3</mn> </mrow> </msup> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <mi>E</mi> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}+F^{n}=(E+F)(E^{n-1}-E^{n-2}F+E^{n-3}F^{2}-\cdots -EF^{n-2}+F^{n-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44bd8709f5595eca7ef48cad2f76b705720ea318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:70.876ex; height:3.176ex;" alt="{\displaystyle E^{n}+F^{n}=(E+F)(E^{n-1}-E^{n-2}F+E^{n-3}F^{2}-\cdots -EF^{n-2}+F^{n-1})}"></span></dd> <dd>(obtained by changing <span class="texhtml mvar" style="font-style:italic;">F</span> by <span class="texhtml">–<i>F</i></span> in the preceding formula)</dd></dl> <ul><li><dl><dt>Sum, even exponent</dt></dl></li></ul> <dl><dd>If the exponent is a power of two then the expression cannot, in general, be factorized without introducing <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex numbers</a> (if <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span> contain complex numbers, this may be not the case). If <i>n</i> has an odd divisor, that is if <span class="texhtml"><i>n</i> = <i>pq</i></span> with <span class="texhtml mvar" style="font-style:italic;">p</span> odd, one may use the preceding formula (in "Sum, odd exponent") applied to <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 (E^{q})^{p}+(F^{q})^{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E^{q})^{p}+(F^{q})^{p}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10a692db0a9e4c9c977fd8c1d5bd51937de791b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.81ex; height:2.843ex;" alt="{\displaystyle (E^{q})^{p}+(F^{q})^{p}.}"></span></dd></dl></dd></dl> <ul><li><dl><dt>Trinomials and cubic formulas</dt></dl></li></ul> <dl><dd><dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&x^{2}+y^{2}+z^{2}+2(xy+yz+xz)=(x+y+z)^{2}\\&x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-xz-yz)\\&x^{3}+y^{3}+z^{3}+3x^{2}(y+z)+3y^{2}(x+z)+3z^{2}(x+y)+6xyz=(x+y+z)^{3}\\&x^{4}+x^{2}y^{2}+y^{4}=(x^{2}+xy+y^{2})(x^{2}-xy+y^{2}).\end{aligned}}}"> <semantics> <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 /> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mi>y</mi> <mi>z</mi> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>3</mn> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mi>z</mi> <mo>−</mo> <mi>y</mi> <mi>z</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>3</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>3</mn> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>6</mn> <mi>x</mi> <mi>y</mi> <mi>z</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <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 stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <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> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&x^{2}+y^{2}+z^{2}+2(xy+yz+xz)=(x+y+z)^{2}\\&x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-xz-yz)\\&x^{3}+y^{3}+z^{3}+3x^{2}(y+z)+3y^{2}(x+z)+3z^{2}(x+y)+6xyz=(x+y+z)^{3}\\&x^{4}+x^{2}y^{2}+y^{4}=(x^{2}+xy+y^{2})(x^{2}-xy+y^{2}).\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/572ec00fc6108808acf9b918447a94022bac2883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.748ex; margin-top: -0.262ex; margin-bottom: -0.256ex; width:75.815ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}&x^{2}+y^{2}+z^{2}+2(xy+yz+xz)=(x+y+z)^{2}\\&x^{3}+y^{3}+z^{3}-3xyz=(x+y+z)(x^{2}+y^{2}+z^{2}-xy-xz-yz)\\&x^{3}+y^{3}+z^{3}+3x^{2}(y+z)+3y^{2}(x+z)+3z^{2}(x+y)+6xyz=(x+y+z)^{3}\\&x^{4}+x^{2}y^{2}+y^{4}=(x^{2}+xy+y^{2})(x^{2}-xy+y^{2}).\end{aligned}}}"></span></dd></dl></dd></dl></dd></dl> <ul><li><dl><dt>Binomial expansions</dt></dl></li></ul> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Binomial_expansion_visualisation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Binomial_expansion_visualisation.svg/300px-Binomial_expansion_visualisation.svg.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Binomial_expansion_visualisation.svg/450px-Binomial_expansion_visualisation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Binomial_expansion_visualisation.svg/600px-Binomial_expansion_visualisation.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>Visualisation of binomial expansion up to the 4th power</figcaption></figure> <dl><dd>The <a href="/wiki/Binomial_theorem" title="Binomial theorem">binomial theorem</a> supplies patterns that can easily be recognized from the integers that appear in them</dd> <dd>In low degree: <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}+2ab+b^{2}=(a+b)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}+2ab+b^{2}=(a+b)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffad2ab3e5599606abd1cbc2e21246a9a2174590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.436ex; height:3.176ex;" alt="{\displaystyle a^{2}+2ab+b^{2}=(a+b)^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{2}-2ab+b^{2}=(a-b)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>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> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{2}-2ab+b^{2}=(a-b)^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36c050369a0ec3e208bb680cb2f1116e02868bc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.436ex; height:3.176ex;" alt="{\displaystyle a^{2}-2ab+b^{2}=(a-b)^{2}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3}}"> <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> <mn>3</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28dc50c53f3e1c2b4eb68f5deb5d4fc4b62b1f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.775ex; height:3.176ex;" alt="{\displaystyle a^{3}+3a^{2}b+3ab^{2}+b^{3}=(a+b)^{3}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{3}-3a^{2}b+3ab^{2}-b^{3}=(a-b)^{3}}"> <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> <mn>3</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>b</mi> <mo>+</mo> <mn>3</mn> <mi>a</mi> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{3}-3a^{2}b+3ab^{2}-b^{3}=(a-b)^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c37d4f1ec52f6646bca194d5d38fba61541154c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.775ex; height:3.176ex;" alt="{\displaystyle a^{3}-3a^{2}b+3ab^{2}-b^{3}=(a-b)^{3}}"></span></dd></dl></dd> <dd>More generally, the coefficients of the expanded forms of <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+b)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+b)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bdfac89abdc81ad9084425d7401403b48d41e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:2.843ex;" alt="{\displaystyle (a+b)^{n}}"></span> and <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-b)^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a-b)^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb72352fcb27938cdf67d6863a6b4781b86711c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.096ex; height:2.843ex;" alt="{\displaystyle (a-b)^{n}}"></span> are the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>, that appear in the <span class="texhtml"><i>n</i></span>th row of <a href="/wiki/Pascal%27s_triangle" title="Pascal's triangle">Pascal's triangle</a>.</dd></dl> <div class="mw-heading mw-heading4"><h4 id="Roots_of_unity">Roots of unity</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=10" title="Edit section: Roots of unity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <span class="texhtml mvar" style="font-style:italic;">n</span>th <a href="/wiki/Roots_of_unity" class="mw-redirect" title="Roots of unity">roots of unity</a> are the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> each of which is a <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> of the polynomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{n}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{n}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8d5639cd7157fe06c622e0509fdec62d5e65e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.198ex; height:2.509ex;" alt="{\displaystyle x^{n}-1.}"></span> They are thus the numbers <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{2ik\pi /n}=\cos {\tfrac {2\pi k}{n}}+i\sin {\tfrac {2\pi k}{n}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>k</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{2ik\pi /n}=\cos {\tfrac {2\pi k}{n}}+i\sin {\tfrac {2\pi k}{n}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69afc27655a4a9e5f0802abe6e383def67199cc5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.094ex; height:3.509ex;" alt="{\displaystyle e^{2ik\pi /n}=\cos {\tfrac {2\pi k}{n}}+i\sin {\tfrac {2\pi k}{n}}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0,\ldots ,n-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k=0,\ldots ,n-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6350249f2bee937389ff8eef95c6e27f7b9f1d47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.695ex; height:2.509ex;" alt="{\displaystyle k=0,\ldots ,n-1.}"></span> </p><p>It follows that for any two expressions <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span>, one has: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}-F^{n}=(E-F)\prod _{k=1}^{n-1}\left(E-Fe^{2ik\pi /n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>−</mo> <mi>F</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>k</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}-F^{n}=(E-F)\prod _{k=1}^{n-1}\left(E-Fe^{2ik\pi /n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbfd576be28eabe8995f8c3053c9b052ef1295c6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:39.455ex; height:7.343ex;" alt="{\displaystyle E^{n}-F^{n}=(E-F)\prod _{k=1}^{n-1}\left(E-Fe^{2ik\pi /n}\right)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}+F^{n}=\prod _{k=0}^{n-1}\left(E-Fe^{(2k+1)i\pi /n}\right)\qquad {\text{if }}n{\text{ is even}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>−</mo> <mi>F</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>i</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is even</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}+F^{n}=\prod _{k=0}^{n-1}\left(E-Fe^{(2k+1)i\pi /n}\right)\qquad {\text{if }}n{\text{ is even}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bca0c880b4ce6ccd128dc8ecdd9322c38432ea02" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.958ex; height:7.509ex;" alt="{\displaystyle E^{n}+F^{n}=\prod _{k=0}^{n-1}\left(E-Fe^{(2k+1)i\pi /n}\right)\qquad {\text{if }}n{\text{ is even}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}+F^{n}=(E+F)\prod _{k=1}^{n-1}\left(E+Fe^{2ik\pi /n}\right)\qquad {\text{if }}n{\text{ is odd}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>+</mo> <mi>F</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>i</mi> <mi>k</mi> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext> is odd</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}+F^{n}=(E+F)\prod _{k=1}^{n-1}\left(E+Fe^{2ik\pi /n}\right)\qquad {\text{if }}n{\text{ is odd}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61de68174ca05dfa9b278069f94d81fe96b07e23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.294ex; height:7.343ex;" alt="{\displaystyle E^{n}+F^{n}=(E+F)\prod _{k=1}^{n-1}\left(E+Fe^{2ik\pi /n}\right)\qquad {\text{if }}n{\text{ is odd}}}"></span> </p><p>If <span class="texhtml mvar" style="font-style:italic;">E</span> and <span class="texhtml mvar" style="font-style:italic;">F</span> are real expressions, and one wants real factors, one has to replace every pair of <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> factors by its product. As the complex conjugate of <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 e^{i\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{i\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dee801d5ef9c81a538f38ecb87dbdc586edae699" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.935ex; height:2.676ex;" alt="{\displaystyle e^{i\alpha }}"></span> is <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 e^{-i\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-i\alpha },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3863041322166dcfdc8bf11fc0f7283ee29ebb48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.86ex; height:3.009ex;" alt="{\displaystyle e^{-i\alpha },}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a-be^{i\alpha }\right)\left(a-be^{-i\alpha }\right)=a^{2}-ab\left(e^{i\alpha }+e^{-i\alpha }\right)+b^{2}e^{i\alpha }e^{-i\alpha }=a^{2}-2ab\cos \,\alpha +b^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>−</mo> <mi>b</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mi>b</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>α</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>α</mi> </mrow> </msup> <mo>=</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> <mi>cos</mi> <mspace width="thinmathspace" /> <mi>α</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a-be^{i\alpha }\right)\left(a-be^{-i\alpha }\right)=a^{2}-ab\left(e^{i\alpha }+e^{-i\alpha }\right)+b^{2}e^{i\alpha }e^{-i\alpha }=a^{2}-2ab\cos \,\alpha +b^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c716f11f50af1fdda6064feb4ccf7d4947736c4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:79.84ex; height:3.343ex;" alt="{\displaystyle \left(a-be^{i\alpha }\right)\left(a-be^{-i\alpha }\right)=a^{2}-ab\left(e^{i\alpha }+e^{-i\alpha }\right)+b^{2}e^{i\alpha }e^{-i\alpha }=a^{2}-2ab\cos \,\alpha +b^{2},}"></span> one has the following real factorizations (one passes from one to the other by changing <span class="texhtml mvar" style="font-style:italic;">k</span> into <span class="texhtml"><i>n</i> – <i>k</i></span> or <span class="texhtml"><i>n</i> + 1 – <i>k</i></span>, and applying the usual <a href="/wiki/Trigonometric_formulas" class="mw-redirect" title="Trigonometric formulas">trigonometric formulas</a>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E^{2n}-F^{2n}&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}-2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\\&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}+2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\end{aligned}}}"> <semantics> <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> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mi>F</mi> <mi>cos</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>−</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>E</mi> <mo>+</mo> <mi>F</mi> <mo stretchy="false">)</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>E</mi> <mi>F</mi> <mi>cos</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mstyle> </mrow> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E^{2n}-F^{2n}&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}-2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\\&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}+2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e41d588991b278c2c35f634bce9cb58c7b80f032" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:60.332ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}E^{2n}-F^{2n}&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}-2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\\&=(E-F)(E+F)\prod _{k=1}^{n-1}\left(E^{2}+2EF\cos \,{\tfrac {k\pi }{n}}+F^{2}\right)\end{aligned}}}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}E^{2n}+F^{2n}&=\prod _{k=1}^{n}\left(E^{2}+2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\\&=\prod _{k=1}^{n}\left(E^{2}-2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\end{aligned}}}"> <semantics> <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> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>E</mi> <mi>F</mi> <mi>cos</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>E</mi> <mi>F</mi> <mi>cos</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mstyle> </mrow> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E^{2n}+F^{2n}&=\prod _{k=1}^{n}\left(E^{2}+2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\\&=\prod _{k=1}^{n}\left(E^{2}-2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ab869f0551b9b93c2109c75bc48503916eb42f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:47.697ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}E^{2n}+F^{2n}&=\prod _{k=1}^{n}\left(E^{2}+2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\\&=\prod _{k=1}^{n}\left(E^{2}-2EF\cos \,{\tfrac {(2k-1)\pi }{2n}}+F^{2}\right)\end{aligned}}}"></span> </p><p>The <a href="/wiki/Cosine" class="mw-redirect" title="Cosine">cosines</a> that appear in these factorizations are <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, and may be expressed in terms of <a href="/wiki/Nth_root" title="Nth root">radicals</a> (this is possible because their <a href="/wiki/Galois_group" title="Galois group">Galois group</a> is cyclic); however, these radical expressions are too complicated to be used, except for low values of <span class="texhtml mvar" style="font-style:italic;">n</span>. For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{4}+b^{4}=(a^{2}-{\sqrt {2}}ab+b^{2})(a^{2}+{\sqrt {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>4</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</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> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <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 stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{4}+b^{4}=(a^{2}-{\sqrt {2}}ab+b^{2})(a^{2}+{\sqrt {2}}ab+b^{2}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1a2d1c1fe1b44cf05f4dd889c5b97a66def70a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.225ex; height:3.176ex;" alt="{\displaystyle a^{4}+b^{4}=(a^{2}-{\sqrt {2}}ab+b^{2})(a^{2}+{\sqrt {2}}ab+b^{2}).}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{5}-b^{5}=(a-b)\left(a^{2}+{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}+{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{5}-b^{5}=(a-b)\left(a^{2}+{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}+{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d77c7b37f34946e4abc9dc25fb1cd5107cd617f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.165ex; height:6.509ex;" alt="{\displaystyle a^{5}-b^{5}=(a-b)\left(a^{2}+{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}+{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{5}+b^{5}=(a+b)\left(a^{2}-{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}-{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>5</mn> </msqrt> </mrow> </mrow> <mn>2</mn> </mfrac> </mrow> <mi>a</mi> <mi>b</mi> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{5}+b^{5}=(a+b)\left(a^{2}-{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}-{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd0910dfd88082a7225a6461f028dde57accf841" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.165ex; height:6.509ex;" alt="{\displaystyle a^{5}+b^{5}=(a+b)\left(a^{2}-{\frac {1-{\sqrt {5}}}{2}}ab+b^{2}\right)\left(a^{2}-{\frac {1+{\sqrt {5}}}{2}}ab+b^{2}\right),}"></span> </p><p>Often one wants a factorization with rational coefficients. Such a factorization involves <a href="/wiki/Cyclotomic_polynomial" title="Cyclotomic polynomial">cyclotomic polynomials</a>. To express rational factorizations of sums and differences or powers, we need a notation for the <a href="/wiki/Homogenization_of_a_polynomial" class="mw-redirect" title="Homogenization of a polynomial">homogenization of a polynomial</a>: if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{0}x^{n}+a_{i}x^{n-1}+\cdots +a_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}x^{n}+a_{i}x^{n-1}+\cdots +a_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130182b9d0d985909d78c8efc4064a2c51848741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.833ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{0}x^{n}+a_{i}x^{n-1}+\cdots +a_{n},}"></span> its <i>homogenization</i> is the <a href="/wiki/Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">bivariate polynomial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {P}}(x,y)=a_{0}x^{n}+a_{i}x^{n-1}y+\cdots +a_{n}y^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>P</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>y</mi> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {P}}(x,y)=a_{0}x^{n}+a_{i}x^{n-1}y+\cdots +a_{n}y^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92731e6e4fbf811d176c094fbb378801cc9838bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.836ex; height:3.509ex;" alt="{\displaystyle {\overline {P}}(x,y)=a_{0}x^{n}+a_{i}x^{n-1}y+\cdots +a_{n}y^{n}.}"></span> Then, one has <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}-F^{n}=\prod _{k\mid n}{\overline {Q}}_{n}(E,F),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∣</mo> <mi>n</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}-F^{n}=\prod _{k\mid n}{\overline {Q}}_{n}(E,F),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4247fb4f5ae19f3563ddd40a360a49efbecadcde" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.519ex; height:6.176ex;" alt="{\displaystyle E^{n}-F^{n}=\prod _{k\mid n}{\overline {Q}}_{n}(E,F),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E^{n}+F^{n}=\prod _{k\mid 2n,k\not \mid n}{\overline {Q}}_{n}(E,F),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msup> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>∣</mo> <mn>2</mn> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>∤</mo> <mi>n</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>E</mi> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E^{n}+F^{n}=\prod _{k\mid 2n,k\not \mid n}{\overline {Q}}_{n}(E,F),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5eb666735323e922f75bd3af07608aaa1ef83ba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:28.429ex; height:6.176ex;" alt="{\displaystyle E^{n}+F^{n}=\prod _{k\mid 2n,k\not \mid n}{\overline {Q}}_{n}(E,F),}"></span> where the products are taken over all divisors of <span class="texhtml mvar" style="font-style:italic;">n</span>, or all divisors of <span class="texhtml">2<i>n</i></span> that do not divide <span class="texhtml mvar" style="font-style:italic;">n</span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q_{n}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q_{n}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5eab5a40e5daf0466f67130cc14c1436a28777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.196ex; height:2.843ex;" alt="{\displaystyle Q_{n}(x)}"></span> is the <span class="texhtml mvar" style="font-style:italic;">n</span>th cyclotomic polynomial. </p><p>For example, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{6}-b^{6}={\overline {Q}}_{1}(a,b){\overline {Q}}_{2}(a,b){\overline {Q}}_{3}(a,b){\overline {Q}}_{6}(a,b)=(a-b)(a+b)(a^{2}-ab+b^{2})(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>6</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <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> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{6}-b^{6}={\overline {Q}}_{1}(a,b){\overline {Q}}_{2}(a,b){\overline {Q}}_{3}(a,b){\overline {Q}}_{6}(a,b)=(a-b)(a+b)(a^{2}-ab+b^{2})(a^{2}+ab+b^{2}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d8ad85718f5890d26a70aa60f468c883a05be80" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:88.193ex; height:3.509ex;" alt="{\displaystyle a^{6}-b^{6}={\overline {Q}}_{1}(a,b){\overline {Q}}_{2}(a,b){\overline {Q}}_{3}(a,b){\overline {Q}}_{6}(a,b)=(a-b)(a+b)(a^{2}-ab+b^{2})(a^{2}+ab+b^{2}),}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a^{6}+b^{6}={\overline {Q}}_{4}(a,b){\overline {Q}}_{12}(a,b)=(a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>Q</mi> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a^{6}+b^{6}={\overline {Q}}_{4}(a,b){\overline {Q}}_{12}(a,b)=(a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff3a825f486a26b997f43cc4c6d86d7c67187cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:56.146ex; height:3.509ex;" alt="{\displaystyle a^{6}+b^{6}={\overline {Q}}_{4}(a,b){\overline {Q}}_{12}(a,b)=(a^{2}+b^{2})(a^{4}-a^{2}b^{2}+b^{4}),}"></span> since the divisors of 6 are 1, 2, 3, 6, and the divisors of 12 that do not divide 6 are 4 and 12. </p> <div class="mw-heading mw-heading2"><h2 id="Polynomials">Polynomials</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=11" title="Edit section: Polynomials"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Factorization_of_polynomials" title="Factorization of polynomials">Factorization of polynomials</a></div> <p>For polynomials, factorization is strongly related with the problem of solving <a href="/wiki/Algebraic_equation" title="Algebraic equation">algebraic equations</a>. An algebraic equation has the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)\ \,{\stackrel {\text{def}}{=}}\ \,a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo>=</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>def</mtext> </mrow> </mover> </mrow> </mrow> <mtext> </mtext> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)\ \,{\stackrel {\text{def}}{=}}\ \,a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c156307c359fbee3cf15d94e4ea3929142f3018" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.333ex; height:3.843ex;" alt="{\displaystyle P(x)\ \,{\stackrel {\text{def}}{=}}\ \,a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n}=0,}"></span></dd></dl> <p>where <span class="texhtml"><i>P</i>(<i>x</i>)</span> is a polynomial in <span class="texhtml mvar" style="font-style:italic;">x</span> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}\neq 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>≠</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}\neq 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/503b981349dab93432690a11cc5ff1cd50d9638e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.192ex; height:2.676ex;" alt="{\displaystyle a_{0}\neq 0.}"></span> A solution of this equation (also called a <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> of the polynomial) is a value <span class="texhtml mvar" style="font-style:italic;">r</span> of <span class="texhtml mvar" style="font-style:italic;">x</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(r)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(r)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f21fd3a8d223a87c7d758fc441aac3e5da94876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.511ex; height:2.843ex;" alt="{\displaystyle P(r)=0.}"></span></dd></dl> <p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=Q(x)R(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=Q(x)R(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5d2fdcca61a7a22b15ed7ed721f3c0f6ed69dea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.863ex; height:2.843ex;" alt="{\displaystyle P(x)=Q(x)R(x)}"></span> is a factorization of <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> as a product of two polynomials, then the roots of <span class="texhtml"><i>P</i>(<i>x</i>)</span> are the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of the roots of <span class="texhtml"><i>Q</i>(<i>x</i>)</span> and the roots of <span class="texhtml"><i>R</i>(<i>x</i>)</span>. Thus solving <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> is reduced to the simpler problems of solving <span class="texhtml"><i>Q</i>(<i>x</i>) = 0</span> and <span class="texhtml"><i>R</i>(<i>x</i>) = 0</span>. </p><p>Conversely, the <a href="/wiki/Factor_theorem" title="Factor theorem">factor theorem</a> asserts that, if <span class="texhtml mvar" style="font-style:italic;">r</span> is a root of <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span>, then <span class="texhtml"><i>P</i>(<i>x</i>)</span> may be factored as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=(x-r)Q(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=(x-r)Q(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d7ac8f6f89d3cd6713acde9678d2afe2968076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.635ex; height:2.843ex;" alt="{\displaystyle P(x)=(x-r)Q(x),}"></span></dd></dl> <p>where <span class="texhtml"><i>Q</i>(<i>x</i>)</span> is the quotient of <a href="/wiki/Euclidean_division_of_polynomials" class="mw-redirect" title="Euclidean division of polynomials">Euclidean division</a> of <span class="texhtml"><i>P</i>(<i>x</i>) = 0</span> by the linear (degree one) factor <span class="texhtml"><i>x</i> – <i>r</i></span>. </p><p>If the coefficients of <span class="texhtml"><i>P</i>(<i>x</i>)</span> are <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex</a> numbers, the <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> asserts that <span class="texhtml"><i>P</i>(<i>x</i>)</span> has a real or complex root. Using the factor theorem recursively, it results that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{0}(x-r_{1})\cdots (x-r_{n}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}(x-r_{1})\cdots (x-r_{n}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e48d17e238c10013e03ef77a8dab5427df3420d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.74ex; height:2.843ex;" alt="{\displaystyle P(x)=a_{0}(x-r_{1})\cdots (x-r_{n}),}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1},\ldots ,r_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1},\ldots ,r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52dfb1691f61905ec0a447728fc01781fcc71205" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.548ex; height:2.009ex;" alt="{\displaystyle r_{1},\ldots ,r_{n}}"></span> are the real or complex roots of <span class="texhtml mvar" style="font-style:italic;">P</span>, with some of them possibly repeated. This complete factorization is unique <a href="/wiki/Up_to" title="Up to">up to</a> the order of the factors. </p><p>If the coefficients of <span class="texhtml"><i>P</i>(<i>x</i>)</span> are real, one generally wants a factorization where factors have real coefficients. In this case, the complete factorization may have some quadratic (degree two) factors. This factorization may easily be deduced from the above complete factorization. In fact, if <span class="texhtml"><i>r</i> = <i>a</i> + <i>ib</i></span> is a non-real root of <span class="texhtml"><i>P</i>(<i>x</i>)</span>, then its <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> <span class="texhtml"><i>s</i> = <i>a</i> - <i>ib</i></span> is also a root of <span class="texhtml"><i>P</i>(<i>x</i>)</span>. So, the product </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-r)(x-s)=x^{2}-(r+s)x+rs=x^{2}-2ax+a^{2}+b^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>+</mo> <mi>r</mi> <mi>s</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>a</mi> <mi>x</mi> <mo>+</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x-r)(x-s)=x^{2}-(r+s)x+rs=x^{2}-2ax+a^{2}+b^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98884b28d624a207f5c2f09ea0093847258fd210" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.58ex; height:3.176ex;" alt="{\displaystyle (x-r)(x-s)=x^{2}-(r+s)x+rs=x^{2}-2ax+a^{2}+b^{2}}"></span></dd></dl> <p>is a factor of <span class="texhtml"><i>P</i>(<i>x</i>)</span> with real coefficients. Repeating this for all non-real factors gives a factorization with linear or quadratic real factors. </p><p>For computing these real or complex factorizations, one needs the roots of the polynomial, which may not be computed exactly, and only approximated using <a href="/wiki/Root-finding_algorithm" title="Root-finding algorithm">root-finding algorithms</a>. </p><p>In practice, most algebraic equations of interest have <a href="/wiki/Integer" title="Integer">integer</a> or <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients, and one may want a factorization with factors of the same kind. The <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a> may be generalized to this case, stating that polynomials with integer or rational coefficients have the <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization property</a>. More precisely, every polynomial with rational coefficients may be factorized in a product </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=q\,P_{1}(x)\cdots P_{k}(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>q</mi> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=q\,P_{1}(x)\cdots P_{k}(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24facbcfd866b8f19c338448800ed0a09cda87c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.989ex; height:2.843ex;" alt="{\displaystyle P(x)=q\,P_{1}(x)\cdots P_{k}(x),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">q</span> is a rational number and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{1}(x),\ldots ,P_{k}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{1}(x),\ldots ,P_{k}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c290a027ff1064b9318dabc2c1f23f45e266f084" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.584ex; height:2.843ex;" alt="{\displaystyle P_{1}(x),\ldots ,P_{k}(x)}"></span> are non-constant polynomials with integer coefficients that are <a href="/wiki/Irreducible_polynomial" title="Irreducible polynomial">irreducible</a> and <a href="/wiki/Primitive_polynomial_(ring_theory)" class="mw-redirect" title="Primitive polynomial (ring theory)">primitive</a>; this means that none of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ff7431d6a65ae949b47316a24ce5b23d9d3ac9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.431ex; height:2.843ex;" alt="{\displaystyle P_{i}(x)}"></span> may be written as the product two polynomials (with integer coefficients) that are neither 1 nor –1 (integers are considered as polynomials of degree zero). Moreover, this factorization is unique up to the order of the factors and the signs of the factors. </p><p>There are efficient <a href="/wiki/Algorithm" title="Algorithm">algorithms</a> for computing this factorization, which are implemented in most <a href="/wiki/Computer_algebra" title="Computer algebra">computer algebra</a> systems. See <a href="/wiki/Factorization_of_polynomials" title="Factorization of polynomials">Factorization of polynomials</a>. Unfortunately, these algorithms are too complicated to use for paper-and-pencil computations. Besides the heuristics above, only a few methods are suitable for hand computations, which generally work only for polynomials of low degree, with few nonzero coefficients. The main such methods are described in next subsections. </p> <div class="mw-heading mw-heading3"><h3 id="Primitive-part_&_content_factorization"><span id="Primitive-part_.26_content_factorization"></span>Primitive-part & content factorization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=12" title="Edit section: Primitive-part & content factorization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polynomial_factorization#Primitive_part–content_factorization" class="mw-redirect" title="Polynomial factorization">Polynomial factorization § Primitive part–content factorization</a></div> <p>Every polynomial with <a href="/wiki/Rational_number" title="Rational number">rational</a> coefficients, may be factorized, in a unique way, as the product of a rational number and a polynomial with integer coefficients, which is <a href="/wiki/Primitive_polynomial_(ring_theory)" class="mw-redirect" title="Primitive polynomial (ring theory)">primitive</a> (that is, the <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">greatest common divisor</a> of the coefficients is 1), and has a positive leading coefficient (coefficient of the term of the highest degree). For example: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -10x^{2}+5x+5=(-5)\cdot (2x^{2}-x-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>10</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <mi>x</mi> <mo>+</mo> <mn>5</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>5</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -10x^{2}+5x+5=(-5)\cdot (2x^{2}-x-1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/762f6856a43d093c56264901fb0f0d3bf033c495" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.938ex; height:3.176ex;" alt="{\displaystyle -10x^{2}+5x+5=(-5)\cdot (2x^{2}-x-1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}x^{5}+{\frac {7}{2}}x^{2}+2x+1={\frac {1}{6}}(2x^{5}+21x^{2}+12x+6)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> <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>1</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> <mo>+</mo> <mn>21</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>12</mn> <mi>x</mi> <mo>+</mo> <mn>6</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}x^{5}+{\frac {7}{2}}x^{2}+2x+1={\frac {1}{6}}(2x^{5}+21x^{2}+12x+6)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c662e602b950b900682346ab253e8c9f5979894b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:49.441ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{3}}x^{5}+{\frac {7}{2}}x^{2}+2x+1={\frac {1}{6}}(2x^{5}+21x^{2}+12x+6)}"></span></dd></dl> <p>In this factorization, the rational number is called the <a href="/wiki/Primitive_part_and_content" title="Primitive part and content">content</a>, and the primitive polynomial is the <a href="/wiki/Primitive_part" class="mw-redirect" title="Primitive part">primitive part</a>. The computation of this factorization may be done as follows: firstly, reduce all coefficients to a common denominator, for getting the quotient by an integer <span class="texhtml mvar" style="font-style:italic;">q</span> of a polynomial with integer coefficients. Then one divides out the greater common divisor <span class="texhtml mvar" style="font-style:italic;">p</span> of the coefficients of this polynomial for getting the primitive part, the content being <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p/q.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p/q.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/859645f5823d17a5abfa11c70bb975118fccebe0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.138ex; height:2.843ex;" alt="{\displaystyle p/q.}"></span> Finally, if needed, one changes the signs of <span class="texhtml mvar" style="font-style:italic;">p</span> and all coefficients of the primitive part. </p><p>This factorization may produce a result that is larger than the original polynomial (typically when there are many <a href="/wiki/Coprime_integers" title="Coprime integers">coprime</a> denominators), but, even when this is the case, the primitive part is generally easier to manipulate for further factorization. </p> <div class="mw-heading mw-heading3"><h3 id="Using_the_factor_theorem">Using the factor theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=13" title="Edit section: Using the factor theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Factor_theorem" title="Factor theorem">Factor theorem</a></div> <p>The factor theorem states that, if <span class="texhtml mvar" style="font-style:italic;">r</span> is a <a href="/wiki/Zero_of_a_function" title="Zero of a function">root</a> of a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f367a967d12fbfcda6fd9a0e22f545c013cb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.806ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}"></span></dd></dl> <p>meaning <span class="texhtml"><i>P</i>(<i>r</i>) = 0</span>, then there is a factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=(x-r)Q(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=(x-r)Q(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d7ac8f6f89d3cd6713acde9678d2afe2968076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.635ex; height:2.843ex;" alt="{\displaystyle P(x)=(x-r)Q(x),}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q(x)=b_{0}x^{n-1}+\cdots +b_{n-2}x+b_{n-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q(x)=b_{0}x^{n-1}+\cdots +b_{n-2}x+b_{n-1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/219630701c49aac0c0aefc92dbdb36c94bc886fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.63ex; height:3.176ex;" alt="{\displaystyle Q(x)=b_{0}x^{n-1}+\cdots +b_{n-2}x+b_{n-1},}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}=b_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}=b_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c244a1fcf1d02c1e6a650c1933c8b1f8d7ecb49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.434ex; height:2.509ex;" alt="{\displaystyle a_{0}=b_{0}}"></span>. Then <a href="/wiki/Polynomial_long_division" title="Polynomial long division">polynomial long division</a> or <a href="/wiki/Synthetic_division" title="Synthetic division">synthetic division</a> give: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{i}=a_{0}r^{i}+\cdots +a_{i-1}r+a_{i}\ {\text{ for }}\ i=1,\ldots ,n{-}1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>r</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mtext> for </mtext> </mrow> <mtext> </mtext> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> </mrow> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b_{i}=a_{0}r^{i}+\cdots +a_{i-1}r+a_{i}\ {\text{ for }}\ i=1,\ldots ,n{-}1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6661ccafdbc6f8a686961d1dfbb9a66129a4f6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.843ex; height:3.009ex;" alt="{\displaystyle b_{i}=a_{0}r^{i}+\cdots +a_{i-1}r+a_{i}\ {\text{ for }}\ i=1,\ldots ,n{-}1.}"></span></dd></dl> <p>This may be useful when one knows or can guess a root of the polynomial. </p><p>For example, for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=x^{3}-3x+2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{3}-3x+2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a62b3fd56ffdae959f0ee266342e490e8f4889" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.349ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{3}-3x+2,}"></span> one may easily see that the sum of its coefficients is 0, so <span class="texhtml"><i>r</i> = 1</span> is a root. As <span class="texhtml"><i>r</i> + 0 = 1</span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r^{2}+0r-3=-2,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0</mn> <mi>r</mi> <mo>−</mo> <mn>3</mn> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r^{2}+0r-3=-2,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55ff313512c1feb7ccd3de7e0f0f1bb632c680d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.873ex; height:3.009ex;" alt="{\displaystyle r^{2}+0r-3=-2,}"></span> one has </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-3x+2=(x-1)(x^{2}+x-2).}"> <semantics> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\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/005822b8fffc656c89c6a0078811430d0811aead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.973ex; height:3.176ex;" alt="{\displaystyle x^{3}-3x+2=(x-1)(x^{2}+x-2).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Rational_roots">Rational roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=14" title="Edit section: Rational roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For polynomials with rational number coefficients, one may search for roots which are rational numbers. Primitive part-content factorization (see <a href="#Primitive_part-content_factorization">above</a>) reduces the problem of searching for rational roots to the case of polynomials with integer coefficients having no non-trivial <a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">common divisor</a>. </p><p>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x={\tfrac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x={\tfrac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe552d231c349d6993c7294b1d42d2c00f5fda4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.091ex; height:3.676ex;" alt="{\displaystyle x={\tfrac {p}{q}}}"></span> is a rational root of such a polynomial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f367a967d12fbfcda6fd9a0e22f545c013cb14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.806ex; height:3.176ex;" alt="{\displaystyle P(x)=a_{0}x^{n}+a_{1}x^{n-1}+\cdots +a_{n-1}x+a_{n},}"></span></dd></dl> <p>the factor theorem shows that one has a factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=(qx-p)Q(x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>q</mi> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=(qx-p)Q(x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e8220b6fdfe56c50344007fdb1a0027418b91a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.825ex; height:2.843ex;" alt="{\displaystyle P(x)=(qx-p)Q(x),}"></span></dd></dl> <p>where both factors have integer coefficients (the fact that <span class="texhtml mvar" style="font-style:italic;">Q</span> has integer coefficients results from the above formula for the quotient of <span class="texhtml"><i>P</i>(<i>x</i>)</span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-p/q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-p/q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a33eba94857dd26c213ec1bd294e31e65a7ec39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.572ex; height:2.843ex;" alt="{\displaystyle x-p/q}"></span>). </p><p>Comparing the coefficients of degree <span class="texhtml mvar" style="font-style:italic;">n</span> and the constant coefficients in the above equality shows that, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b38d2684323653daafdd152b7e988594003897d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.663ex; height:3.676ex;" alt="{\displaystyle {\tfrac {p}{q}}}"></span> is a rational root in <a href="/wiki/Reduced_fraction" class="mw-redirect" title="Reduced fraction">reduced form</a>, then <span class="texhtml mvar" style="font-style:italic;">q</span> is a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b05c0ff6c1f32fc92b5f63a1bd10fe6eb2ab1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.931ex; height:2.009ex;" alt="{\displaystyle a_{0},}"></span> and <span class="texhtml mvar" style="font-style:italic;">p</span> is a divisor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb4733ab60320bf912d93dce5a27ccd1dca7e905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.095ex; height:2.009ex;" alt="{\displaystyle a_{n}.}"></span> Therefore, there is a finite number of possibilities for <span class="texhtml mvar" style="font-style:italic;">p</span> and <span class="texhtml mvar" style="font-style:italic;">q</span>, which can be systematically examined.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>For example, if the polynomial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=2x^{3}-7x^{2}+10x-6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)=2x^{3}-7x^{2}+10x-6}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a6d11d20c6436d2c501e565bc29eafda04e4db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.414ex; height:3.176ex;" alt="{\displaystyle P(x)=2x^{3}-7x^{2}+10x-6}"></span></dd></dl> <p>has a rational root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {p}{q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{q}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b38d2684323653daafdd152b7e988594003897d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:1.663ex; height:3.676ex;" alt="{\displaystyle {\tfrac {p}{q}}}"></span> with <span class="texhtml"><i>q</i> > 0</span>, then <span class="texhtml mvar" style="font-style:italic;">p</span> must divide 6; that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in \{\pm 1,\pm 2,\pm 3,\pm 6\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>±</mo> <mn>3</mn> <mo>,</mo> <mo>±</mo> <mn>6</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\in \{\pm 1,\pm 2,\pm 3,\pm 6\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34877f18a54a5357158259a52a80a0a5956e6654" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:22.055ex; height:2.843ex;" alt="{\displaystyle p\in \{\pm 1,\pm 2,\pm 3,\pm 6\},}"></span> and <span class="texhtml mvar" style="font-style:italic;">q</span> must divide 2, that is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\in \{1,2\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\in \{1,2\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4584ea0eab877279743c10dd48b9ea500cad5db5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.241ex; height:2.843ex;" alt="{\displaystyle q\in \{1,2\}.}"></span> Moreover, if <span class="texhtml"><i>x</i> < 0</span>, all terms of the polynomial are negative, and, therefore, a root cannot be negative. That is, one must have </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {p}{q}}\in \{1,2,3,6,{\tfrac {1}{2}},{\tfrac {3}{2}}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mstyle> </mrow> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {p}{q}}\in \{1,2,3,6,{\tfrac {1}{2}},{\tfrac {3}{2}}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6d0e81d3e78868272f3199e46c9b35d1eb236bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.611ex; height:3.676ex;" alt="{\displaystyle {\tfrac {p}{q}}\in \{1,2,3,6,{\tfrac {1}{2}},{\tfrac {3}{2}}\}.}"></span></dd></dl> <p>A direct computation shows that only <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {3}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {3}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/631d66184353d37ebfe470a07a6a61487da227ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {3}{2}}}"></span> is a root, so there can be no other rational root. Applying the factor theorem leads finally to the factorization <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2x^{3}-7x^{2}+10x-6=(2x-3)(x^{2}-2x+2).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>7</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>10</mn> <mi>x</mi> <mo>−</mo> <mn>6</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</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>2</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2x^{3}-7x^{2}+10x-6=(2x-3)(x^{2}-2x+2).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a81b9c155352f1b268ee429ddef9d17f3419d3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.009ex; height:3.176ex;" alt="{\displaystyle 2x^{3}-7x^{2}+10x-6=(2x-3)(x^{2}-2x+2).}"></span> </p> <div class="mw-heading mw-heading4"><h4 id="Quadratic_ac_method">Quadratic ac method</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=15" title="Edit section: Quadratic ac method"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The above method may be adapted for <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomials</a>, leading to the <i>ac method</i> of factorization.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Consider the quadratic polynomial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=ax^{2}+bx+c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>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> <annotation encoding="application/x-tex">{\displaystyle P(x)=ax^{2}+bx+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8fb6c8d02a6e30ee4263a2f4a30799fb78f988f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.611ex; height:3.176ex;" alt="{\displaystyle P(x)=ax^{2}+bx+c}"></span></dd></dl> <p>with integer coefficients. If it has a rational root, its denominator must divide <span class="texhtml"><i>a</i></span> evenly and it may be written as a possibly <a href="/wiki/Irreducible_fraction" title="Irreducible fraction">reducible fraction</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}={\tfrac {r}{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}={\tfrac {r}{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d753c0165ba41fc56a42a9f86cd359ef72b6d550" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.554ex; height:3.009ex;" alt="{\displaystyle r_{1}={\tfrac {r}{a}}.}"></span> By <a href="/wiki/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formulas</a>, the other root <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbe9b0b294fdd6fadbf9a7249813f016dcbc44f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{2}}"></span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{2}=-{\frac {b}{a}}-r_{1}=-{\frac {b}{a}}-{\frac {r}{a}}=-{\frac {b+r}{a}}={\frac {s}{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>b</mi> <mi>a</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>b</mi> <mo>+</mo> <mi>r</mi> </mrow> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{2}=-{\frac {b}{a}}-r_{1}=-{\frac {b}{a}}-{\frac {r}{a}}=-{\frac {b+r}{a}}={\frac {s}{a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0adeb6e7d75dca21815132bfefadc16135b4a79c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:42.338ex; height:5.343ex;" alt="{\displaystyle r_{2}=-{\frac {b}{a}}-r_{1}=-{\frac {b}{a}}-{\frac {r}{a}}=-{\frac {b+r}{a}}={\frac {s}{a}},}"></span></dd></dl> <p>with <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 s=-(b+r).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=-(b+r).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b150867227de542088bd5d35f42aade276e198c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.34ex; height:2.843ex;" alt="{\displaystyle s=-(b+r).}"></span> Thus the second root is also rational, and Vieta's second formula <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}r_{2}={\frac {c}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}r_{2}={\frac {c}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e35fa4972106303c13a02cb2c6038226b2ac848" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.37ex; height:4.676ex;" alt="{\displaystyle r_{1}r_{2}={\frac {c}{a}}}"></span> gives </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {s}{a}}{\frac {r}{a}}={\frac {c}{a}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mi>a</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {s}{a}}{\frac {r}{a}}={\frac {c}{a}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa15de232f42a0ea882e56d91341ee5ca769579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.943ex; height:4.676ex;" alt="{\displaystyle {\frac {s}{a}}{\frac {r}{a}}={\frac {c}{a}},}"></span></dd></dl> <p>that is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rs=ac\quad {\text{and}}\quad r+s=-b.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>s</mi> <mo>=</mo> <mi>a</mi> <mi>c</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>b</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rs=ac\quad {\text{and}}\quad r+s=-b.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf318ad29b70293d3638b92b5434460aef2a203" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:27.397ex; height:2.343ex;" alt="{\displaystyle rs=ac\quad {\text{and}}\quad r+s=-b.}"></span></dd></dl> <p>Checking all pairs of integers whose product is <span class="texhtml"><i>ac</i></span> gives the rational roots, if any. </p><p>In summary, if <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 ax^{2}+bx+c}"> <semantics> <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> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126c6935d3dd9f1c1da0c388ca2799be4f6f237c" 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 ax^{2}+bx+c}"></span> has rational roots there are integers <span class="texhtml mvar" style="font-style:italic;">r</span> and <span class="texhtml mvar" style="font-style:italic;">s</span> such <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 rs=ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>s</mi> <mo>=</mo> <mi>a</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rs=ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87217aab59aec8f9e4441a09c4d7f7aa46301719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.474ex; height:1.676ex;" alt="{\displaystyle rs=ac}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r+s=-b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo>=</mo> <mo>−</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r+s=-b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4107118abc75f5d646baabc61f947b4ae311e21f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.884ex; height:2.343ex;" alt="{\displaystyle r+s=-b}"></span> (a finite number of cases to test), and the roots are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {r}{a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>r</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {r}{a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/090bc5c6eac1ba2a182e85ffe368c576ef372f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.706ex; height:3.009ex;" alt="{\displaystyle {\tfrac {r}{a}}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {s}{a}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>s</mi> <mi>a</mi> </mfrac> </mstyle> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {s}{a}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e605a87cfecb2dd5626eaee9f79fa48cea846016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.353ex; height:3.009ex;" alt="{\displaystyle {\tfrac {s}{a}}.}"></span> In other words, one has the factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(ax^{2}+bx+c)=(ax-r)(ax-s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <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 stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>−</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>x</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(ax^{2}+bx+c)=(ax-r)(ax-s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0843e39035b23d977fec1ddcd54e88319171c77c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.97ex; height:3.176ex;" alt="{\displaystyle a(ax^{2}+bx+c)=(ax-r)(ax-s).}"></span></dd></dl> <p>For example, let consider the quadratic polynomial </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{2}+13x+6.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>13</mn> <mi>x</mi> <mo>+</mo> <mn>6.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6x^{2}+13x+6.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c43d377aefad78a7c34ac7bb1a9ffa256bb0e1e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.691ex; height:2.843ex;" alt="{\displaystyle 6x^{2}+13x+6.}"></span></dd></dl> <p>Inspection of the factors of <span class="texhtml"><i>ac</i> = 36</span> leads to <span class="texhtml">4 + 9 = 13 = <i>b</i></span>, giving the two roots </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}=-{\frac {4}{6}}=-{\frac {2}{3}}\quad {\text{and}}\quad r_{2}=-{\frac {9}{6}}=-{\frac {3}{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>6</mn> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}=-{\frac {4}{6}}=-{\frac {2}{3}}\quad {\text{and}}\quad r_{2}=-{\frac {9}{6}}=-{\frac {3}{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0411c4ac7f43659c36b8b3c04e12c4b65393fe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.866ex; height:5.176ex;" alt="{\displaystyle r_{1}=-{\frac {4}{6}}=-{\frac {2}{3}}\quad {\text{and}}\quad r_{2}=-{\frac {9}{6}}=-{\frac {3}{2}},}"></span></dd></dl> <p>and the factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6x^{2}+13x+6=6(x+{\tfrac {2}{3}})(x+{\tfrac {3}{2}})=(3x+2)(2x+3).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>13</mn> <mi>x</mi> <mo>+</mo> <mn>6</mn> <mo>=</mo> <mn>6</mn> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6x^{2}+13x+6=6(x+{\tfrac {2}{3}})(x+{\tfrac {3}{2}})=(3x+2)(2x+3).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/754b767fb7ee0f024ffbf8793acfa76122c15029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:53.934ex; height:3.676ex;" alt="{\displaystyle 6x^{2}+13x+6=6(x+{\tfrac {2}{3}})(x+{\tfrac {3}{2}})=(3x+2)(2x+3).}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Using_formulas_for_polynomial_roots">Using formulas for polynomial roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=16" title="Edit section: Using formulas for polynomial roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any univariate <a href="/wiki/Quadratic_polynomial" class="mw-redirect" title="Quadratic polynomial">quadratic polynomial</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c}"> <semantics> <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> <annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126c6935d3dd9f1c1da0c388ca2799be4f6f237c" 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 ax^{2}+bx+c}"></span> can be factored using the <a href="/wiki/Quadratic_formula" title="Quadratic formula">quadratic formula</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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"> <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> <annotation encoding="application/x-tex">{\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/c54c00fec952fe587b81a734554d609131ae9fba" 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 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></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" 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 \beta }"></span> are the two <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> of the polynomial. </p><p>If <span class="texhtml"><i>a, b, c</i></span> are all <a href="/wiki/Real_number" title="Real number">real</a>, the factors are real if and only if the <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b^{2}-4ac}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b^{2}-4ac}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2c88a48e0087a5786b460b2e856d118b5e23ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.291ex; height:2.843ex;" alt="{\displaystyle b^{2}-4ac}"></span> is non-negative. Otherwise, the quadratic polynomial cannot be factorized into non-constant real factors. </p><p>The quadratic formula is valid when the coefficients belong to any <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> different from two, and, in particular, for coefficients in a <a href="/wiki/Finite_field" title="Finite field">finite field</a> with an odd number of elements.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>There are also formulas for roots of <a href="/wiki/Cubic_function" title="Cubic function">cubic</a> and <a href="/wiki/Quartic_function" title="Quartic function">quartic</a> polynomials, which are, in general, too complicated for practical use. The <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> shows that there are no general root formulas in terms of radicals for polynomials of degree five or higher. </p> <div class="mw-heading mw-heading3"><h3 id="Using_relations_between_roots">Using relations between roots</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=17" title="Edit section: Using relations between roots"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It may occur that one knows some relationship between the roots of a polynomial and its coefficients. Using this knowledge may help factoring the polynomial and finding its roots. <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> is based on a systematic study of the relations between roots and coefficients, that include <a href="/wiki/Vieta%27s_formulas" title="Vieta's formulas">Vieta's formulas</a>. </p><p>Here, we consider the simpler case where two roots <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7af1b928f06e4c7e3e8ebfd60704656719bd766" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}"></span> of a polynomial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"></span> satisfy the relation </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}=Q(x_{1}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{2}=Q(x_{1}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19c9a5fde5ba2b55966fa0e2aa217a30f9203593" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.161ex; height:2.843ex;" alt="{\displaystyle x_{2}=Q(x_{1}),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">Q</span> is a polynomial. </p><p>This implies that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8788bf85d532fa88d1fb25eff6ae382a601c308" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}"></span> is a common root of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(Q(x))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(Q(x))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9068710c978afd19c7c10b275b1c2c652261a0a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.532ex; height:2.843ex;" alt="{\displaystyle P(Q(x))}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb95461f16ba60ea4c17f47816a028e2fe113dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:2.843ex;" alt="{\displaystyle P(x).}"></span> It is therefore a root of the <a href="/wiki/Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">greatest common divisor</a> of these two polynomials. It follows that this greatest common divisor is a non constant factor of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb95461f16ba60ea4c17f47816a028e2fe113dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:2.843ex;" alt="{\displaystyle P(x).}"></span> <a href="/wiki/Euclidean_algorithm_for_polynomials" class="mw-redirect" title="Euclidean algorithm for polynomials">Euclidean algorithm for polynomials</a> allows computing this greatest common factor. </p><p>For example,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> if one know or guess that: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)=x^{3}-5x^{2}-16x+80}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>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> <annotation encoding="application/x-tex">{\displaystyle P(x)=x^{3}-5x^{2}-16x+80}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ebfd464bd61da46cdb9d18c82c8aedfe59dac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.414ex; height:3.176ex;" alt="{\displaystyle P(x)=x^{3}-5x^{2}-16x+80}"></span> has two roots that sum to zero, one may apply Euclidean algorithm to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(-x).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(-x).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dfd97a65098c2cb73a6c277561f8db7064ef0b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.339ex; height:2.843ex;" alt="{\displaystyle P(-x).}"></span> The first division step consists in adding <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"></span> to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(-x),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(-x),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46587f5d1a59a149bcd580dee3eb09584bc3205a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.339ex; height:2.843ex;" alt="{\displaystyle P(-x),}"></span> giving the <a href="/wiki/Remainder" title="Remainder">remainder</a> of </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -10(x^{2}-16).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mn>10</mn> <mo stretchy="false">(</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>16</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -10(x^{2}-16).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b9b20f98f02286a8893afad71117db1cc7b5bb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.138ex; height:3.176ex;" alt="{\displaystyle -10(x^{2}-16).}"></span></dd></dl> <p>Then, dividing <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89833156eff2c51bfb8750db3306a0544ce34e14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.884ex; height:2.843ex;" alt="{\displaystyle P(x)}"></span> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}-16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}-16}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b2e71d39f209875efddc3e6d93a2d6815c71d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.549ex; height:2.843ex;" alt="{\displaystyle x^{2}-16}"></span> gives zero as a new remainder, and <span class="texhtml"><i>x</i> – 5</span> as a quotient, leading to the complete factorization </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).}"> <semantics> <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> <annotation encoding="application/x-tex">{\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/f1253c43f8c1eebd09e41765bb286c510488f17d" 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 x^{3}-5x^{2}-16x+80=(x-5)(x-4)(x+4).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Unique_factorization_domains">Unique factorization domains</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=18" title="Edit section: Unique factorization domains"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The integers and the polynomials over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> share the property of unique factorization, that is, every nonzero element may be factored into a product of an invertible element (a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>, ±1 in the case of integers) and a product of <a href="/wiki/Irreducible_element" title="Irreducible element">irreducible elements</a> (<a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, in the case of integers), and this factorization is unique up to rearranging the factors and shifting units among the factors. <a href="/wiki/Integral_domain" title="Integral domain">Integral domains</a> which share this property are called <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a> (UFD). </p><p><a href="/wiki/Greatest_common_divisor" title="Greatest common divisor">Greatest common divisors</a> exist in UFDs, but not every integral domain in which greatest common divisors exist (known as a <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a>) is a UFD. Every <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a> is a UFD. </p><p>A <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a> is an integral domain on which is defined a <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a> similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. </p><p>In a Euclidean domain, Euclidean division allows defining a <a href="/wiki/Euclidean_algorithm" title="Euclidean algorithm">Euclidean algorithm</a> for computing greatest common divisors. However this does not imply the existence of a factorization algorithm. There is an explicit example of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <span class="texhtml mvar" style="font-style:italic;">F</span> such that there cannot exist any factorization algorithm in the Euclidean domain <span class="texhtml"><i>F</i>[<i>x</i>]</span> of the univariate polynomials over <span class="texhtml mvar" style="font-style:italic;">F</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Ideals">Ideals</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=19" title="Edit section: Ideals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a></div> <p>In <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>, the study of <a href="/wiki/Diophantine_equation" title="Diophantine equation">Diophantine equations</a> led mathematicians, during 19th century, to introduce generalizations of the <a href="/wiki/Integer" title="Integer">integers</a> called <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>. The first <a href="/wiki/Ring_of_algebraic_integers" class="mw-redirect" title="Ring of algebraic integers">ring of algebraic integers</a> that have been considered were <a href="/wiki/Gaussian_integer" title="Gaussian integer">Gaussian integers</a> and <a href="/wiki/Eisenstein_integer" title="Eisenstein integer">Eisenstein integers</a>, which share with usual integers the property of being <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a>, and have thus the <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization property</a>. </p><p>Unfortunately, it soon appeared that most rings of algebraic integers are not principal and do not have unique factorization. The simplest example is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [{\sqrt {-5}}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [{\sqrt {-5}}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643ef2c1e3b8b8e684a0ddf73d649cb0202cc0c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.397ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} [{\sqrt {-5}}],}"></span> in which </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 9=3\cdot 3=(2+{\sqrt {-5}})(2-{\sqrt {-5}}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>9</mn> <mo>=</mo> <mn>3</mn> <mo>⋅</mo> <mn>3</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>5</mn> </msqrt> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 9=3\cdot 3=(2+{\sqrt {-5}})(2-{\sqrt {-5}}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebcfa3483249d475111d8b2b60183afd5a298eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.447ex; height:3.009ex;" alt="{\displaystyle 9=3\cdot 3=(2+{\sqrt {-5}})(2-{\sqrt {-5}}),}"></span></dd></dl> <p>and all these factors are <a href="/wiki/Irreducible_element" title="Irreducible element">irreducible</a>. </p><p>This lack of unique factorization is a major difficulty for solving Diophantine equations. For example, many wrong proofs of <a href="/wiki/Fermat%27s_Last_Theorem" title="Fermat's Last Theorem">Fermat's Last Theorem</a> (probably including <a href="/wiki/Pierre_de_Fermat" title="Pierre de Fermat">Fermat's</a> <i>"truly marvelous proof of this, which this margin is too narrow to contain"</i>) were based on the implicit supposition of unique factorization. </p><p>This difficulty was resolved by <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a>, who proved that the rings of algebraic integers have unique factorization of <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a>: in these rings, every ideal is a product of <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a>, and this factorization is unique up the order of the factors. The <a href="/wiki/Integral_domain" title="Integral domain">integral domains</a> that have this unique factorization property are now called <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a>. They have many nice properties that make them fundamental in algebraic number theory. </p> <div class="mw-heading mw-heading2"><h2 id="Matrices">Matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=20" title="Edit section: Matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Matrix rings are non-commutative and have no unique factorization: there are, in general, many ways of writing a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> as a product of matrices. Thus, the factorization problem consists of finding factors of specified types. For example, the <a href="/wiki/LU_decomposition" title="LU decomposition">LU decomposition</a> gives a matrix as the product of a <a href="/wiki/Lower_triangular_matrix" class="mw-redirect" title="Lower triangular matrix">lower triangular matrix</a> by an <a href="/wiki/Upper_triangular_matrix" class="mw-redirect" title="Upper triangular matrix">upper triangular matrix</a>. As this is not always possible, one generally considers the "LUP decomposition" having a <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a> as its third factor. </p><p>See <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Matrix decomposition</a> for the most common types of matrix factorizations. </p><p>A <a href="/wiki/Logical_matrix" title="Logical matrix">logical matrix</a> represents a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a>, and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> corresponds to <a href="/wiki/Composition_of_relations" title="Composition of relations">composition of relations</a>. Decomposition of a relation through factorization serves to profile the nature of the relation, such as a <a href="/wiki/Difunctional" class="mw-redirect" title="Difunctional">difunctional</a> relation. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=21" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Euler%27s_factorization_method" title="Euler's factorization method">Euler's factorization method</a> for integers</li> <li><a href="/wiki/Fermat%27s_factorization_method" title="Fermat's factorization method">Fermat's factorization method</a> for integers</li> <li><a href="/wiki/Monoid_factorisation" title="Monoid factorisation">Monoid factorisation</a></li> <li><a href="/wiki/Multiplicative_partition" title="Multiplicative partition">Multiplicative partition</a></li> <li><a href="/wiki/Table_of_Gaussian_integer_factorizations" title="Table of Gaussian integer factorizations">Table of Gaussian integer factorizations</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=22" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHardyWright1980" class="citation book cs2">Hardy; Wright (1980), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductiontoth00hard"><i>An Introduction to the Theory of Numbers</i></a></span> (5th ed.), Oxford Science Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0198531715" title="Special:BookSources/978-0198531715"><bdi>978-0198531715</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Numbers&rft.edition=5th&rft.pub=Oxford+Science+Publications&rft.date=1980&rft.isbn=978-0198531715&rft.au=Hardy&rft.au=Wright&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontoth00hard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFKlein1925">Klein 1925</a>, pp. 101–102</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">In <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSanford2008" class="citation cs2">Sanford, Vera (2008) [1930], <i>A Short History of Mathematics</i>, Read Books, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781409727101" title="Special:BookSources/9781409727101"><bdi>9781409727101</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Short+History+of+Mathematics&rft.pub=Read+Books&rft.date=2008&rft.isbn=9781409727101&rft.aulast=Sanford&rft.aufirst=Vera&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span>, the author notes "In view of the present emphasis given to the solution of quadratic equations by factoring, it is interesting to note that this method was not used until Harriot's work of 1631".</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarriot1631" class="citation book cs2 cs1-prop-foreign-lang-source">Harriot, T. (1631), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=771CAAAAcAAJ"><i>Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas</i></a> (in Latin), Apud Robertum Barker, typographum regium</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Artis+Analyticae+Praxis+ad+Aequationes+Algebraicas+Resolvendas&rft.pub=Apud+Robertum+Barker%2C+typographum+regium&rft.date=1631&rft.aulast=Harriot&rft.aufirst=T.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D771CAAAAcAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFFite1921">Fite 1921</a>, p. 19</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFSelby1970">Selby 1970</a>, p. 101</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFDickson1922">Dickson 1922</a>, p. 27</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStover" class="citation web cs2">Stover, Christopher, <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ACMethod.html">"AC Method"</a>, <i>Mathworld</i>, <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141112231252/http://mathworld.wolfram.com/ACMethod.html">archived</a> from the original on 2014-11-12</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Mathworld&rft.atitle=AC+Method&rft.aulast=Stover&rft.aufirst=Christopher&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FACMethod.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">In a field of characteristic 2, one has 2 = 0, and the formula produces a division by zero.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFBurnsidePanton1960">Burnside & Panton 1960</a>, p. 38</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=23" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBurnsidePanton1960" class="citation cs2"><a href="/wiki/William_Burnside" title="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 one)</i>, Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Theory+of+Equations+with+an+introduction+to+the+theory+of+binary+algebraic+forms+%28Volume+one%29&rft.pub=Dover&rft.date=1960&rft.aulast=Burnside&rft.aufirst=William+Snow&rft.au=Panton%2C+Arthur+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDickson1922" class="citation cs2"><a href="/wiki/Leonard_Eugene_Dickson" title="Leonard Eugene Dickson">Dickson, Leonard Eugene</a> (1922), <i>First Course in the Theory of Equations</i>, New York: John Wiley & Sons</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=First+Course+in+the+Theory+of+Equations&rft.place=New+York&rft.pub=John+Wiley+%26+Sons&rft.date=1922&rft.aulast=Dickson&rft.aufirst=Leonard+Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFite1921" class="citation cs2">Fite, William Benjamin (1921), <i>College Algebra (Revised)</i>, Boston: D. C. Heath & Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=College+Algebra+%28Revised%29&rft.place=Boston&rft.pub=D.+C.+Heath+%26+Co.&rft.date=1921&rft.aulast=Fite&rft.aufirst=William+Benjamin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKlein1925" class="citation cs2"><a href="/wiki/Felix_Klein" title="Felix Klein">Klein, Felix</a> (1925), <i>Elementary Mathematics from an Advanced Standpoint; Arithmetic, Algebra, Analysis</i>, Dover</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Mathematics+from+an+Advanced+Standpoint%3B+Arithmetic%2C+Algebra%2C+Analysis&rft.pub=Dover&rft.date=1925&rft.aulast=Klein&rft.aufirst=Felix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSelby1970" class="citation cs2">Selby, Samuel M. (1970), <i>CRC Standard Mathematical Tables</i> (18th ed.), The Chemical Rubber Co.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=CRC+Standard+Mathematical+Tables&rft.edition=18th&rft.pub=The+Chemical+Rubber+Co.&rft.date=1970&rft.aulast=Selby&rft.aufirst=Samuel+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFactorization" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Factorization&action=edit&section=24" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid 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screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/40px-Wiktionary-logo-en-v2.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/60px-Wiktionary-logo-en-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Wiktionary-logo-en-v2.svg/80px-Wiktionary-logo-en-v2.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span></div> <div class="side-box-text plainlist">Look up <i><b><a href="https://en.wiktionary.org/wiki/factorisation" class="extiw" title="wiktionary:factorisation">factorisation</a></b></i> or <i><b><a href="https://en.wiktionary.org/wiki/factorization" class="extiw" title="wiktionary:factorization">factorization</a></b></i> in Wiktionary, the free dictionary.</div></div> </div> <ul><li><a href="/wiki/Wolfram_Alpha" class="mw-redirect" title="Wolfram Alpha">Wolfram Alpha</a> <a rel="nofollow" class="external text" href="http://www.wolframalpha.com/input/?i=Factor%20-2006+%2B+1155+x+-+78+x^2+%2B+x^3">can factorize too</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist 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Factorization - Wikipedia