Diskriminanta - Wikipedija, prosta enciklopedija

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class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="//donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_sl.wikipedia.org&uselang=sl" class=""><span>Denarni prispevki</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Registracija&returnto=Diskriminanta" title="Predlagamo vam, da si ustvarite račun in se prijavite, vendar to ni obvezno." class=""><span>Ustvari račun</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Posebno:Prijava&returnto=Diskriminanta" title="Prijava je zaželena, vendar ni obvezna [o]" accesskey="o" class=""><span>Prijava</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Več možnosti" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Osebna orodja" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Osebna orodja</span> </label> <div 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<div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Projekt"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Vsebina" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Vsebina</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">prestavi v stransko letvico</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrij</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Uvod</div> </a> </li> <li id="toc-Definicija" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definicija"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definicija</span> </div> </a> <ul id="toc-Definicija-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Posplošitev" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Posplošitev"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Posplošitev</span> </div> </a> <ul id="toc-Posplošitev-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diskriminanta_mnogočlenika" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diskriminanta_mnogočlenika"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Diskriminanta mnogočlenika</span> </div> </a> <button aria-controls="toc-Diskriminanta_mnogočlenika-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>Vklopi podrazdelek Diskriminanta mnogočlenika</span> </button> <ul id="toc-Diskriminanta_mnogočlenika-sublist" class="vector-toc-list"> <li id="toc-Zgled" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zgled"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Zgled</span> </div> </a> <ul id="toc-Zgled-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Lastnosti_ničel" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Lastnosti_ničel"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Lastnosti ničel</span> </div> </a> <button aria-controls="toc-Lastnosti_ničel-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>Vklopi podrazdelek Lastnosti ničel</span> </button> <ul id="toc-Lastnosti_ničel-sublist" class="vector-toc-list"> <li id="toc-Kvadratni_mnogočleniki" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kvadratni_mnogočleniki"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Kvadratni mnogočleniki</span> </div> </a> <ul id="toc-Kvadratni_mnogočleniki-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Kubični_mnogočlenik" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Kubični_mnogočlenik"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Kubični mnogočlenik</span> </div> </a> <ul id="toc-Kubični_mnogočlenik-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diskriminanta_stožnic" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diskriminanta_stožnic"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Diskriminanta stožnic</span> </div> </a> <ul id="toc-Diskriminanta_stožnic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Opombe_in_sklici" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Opombe_in_sklici"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Opombe in sklici</span> </div> </a> <ul id="toc-Opombe_in_sklici-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zunanje_povezave" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Zunanje_povezave"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Zunanje povezave</span> </div> </a> <ul id="toc-Zunanje_povezave-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="Vsebina" 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="Vklopi kazalo vsebine" > <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">Vklopi kazalo vsebine</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">Diskriminanta</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="P9jdi na članek v drugem jeziku. Na voljo v 43 jezikih." > <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-43" 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">43 jezikov</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%85%D9%8A%D8%B2" title="مميز – arabščina" lang="ar" hreflang="ar" data-title="مميز" data-language-autonym="العربية" data-language-local-name="arabščina" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%81%D0%BA%D1%80%D1%8B%D0%BC%D1%96%D0%BD%D0%B0%D0%BD%D1%82" title="Дыскрымінант – beloruščina" lang="be" hreflang="be" data-title="Дыскрымінант" data-language-autonym="Беларуская" data-language-local-name="beloruščina" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82%D0%B0" title="Дискриминанта – bolgarščina" lang="bg" hreflang="bg" data-title="Дискриминанта" data-language-autonym="Български" data-language-local-name="bolgarščina" 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/Discriminant" title="Discriminant – katalonščina" lang="ca" hreflang="ca" data-title="Discriminant" data-language-autonym="Català" data-language-local-name="katalonščina" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Diskriminant" title="Diskriminant – češčina" lang="cs" hreflang="cs" data-title="Diskriminant" data-language-autonym="Čeština" data-language-local-name="češčina" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант – čuvaščina" lang="cv" hreflang="cv" data-title="Дискриминант" data-language-autonym="Чӑвашла" data-language-local-name="čuvaščina" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diskriminant" title="Diskriminant – danščina" lang="da" hreflang="da" data-title="Diskriminant" data-language-autonym="Dansk" data-language-local-name="danščina" 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/Diskriminante" title="Diskriminante – nemščina" lang="de" hreflang="de" data-title="Diskriminante" data-language-autonym="Deutsch" data-language-local-name="nemščina" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Discriminant" title="Discriminant – angleščina" lang="en" hreflang="en" data-title="Discriminant" data-language-autonym="English" data-language-local-name="angleščina" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Diskriminanto" title="Diskriminanto – esperanto" lang="eo" hreflang="eo" data-title="Diskriminanto" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Discriminante" title="Discriminante – španščina" lang="es" hreflang="es" data-title="Discriminante" data-language-autonym="Español" data-language-local-name="španščina" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A8%DB%8C%D9%86_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="مبین (ریاضیات) – perzijščina" lang="fa" hreflang="fa" data-title="مبین (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="perzijščina" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diskriminantti" title="Diskriminantti – finščina" lang="fi" hreflang="fi" data-title="Diskriminantti" data-language-autonym="Suomi" data-language-local-name="finščina" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Discriminant" title="Discriminant – francoščina" lang="fr" hreflang="fr" data-title="Discriminant" data-language-autonym="Français" data-language-local-name="francoščina" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Idirdheala%C3%AD" title="Idirdhealaí – irščina" lang="ga" hreflang="ga" data-title="Idirdhealaí" data-language-autonym="Gaeilge" data-language-local-name="irščina" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%93%D7%99%D7%A1%D7%A7%D7%A8%D7%99%D7%9E%D7%99%D7%A0%D7%A0%D7%98%D7%94" title="דיסקרימיננטה – hebrejščina" lang="he" hreflang="he" data-title="דיסקרימיננטה" data-language-autonym="עברית" data-language-local-name="hebrejščina" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diszkrimin%C3%A1ns" title="Diszkrimináns – madžarščina" lang="hu" hreflang="hu" data-title="Diszkrimináns" data-language-autonym="Magyar" data-language-local-name="madžarščina" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BD%D5%AF%D6%80%D5%AB%D5%B4%D5%AB%D5%B6%D5%A1%D5%B6%D5%BF" title="Դիսկրիմինանտ – armenščina" lang="hy" hreflang="hy" data-title="Դիսկրիմինանտ" data-language-autonym="Հայերեն" data-language-local-name="armenščina" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Diskriminan" title="Diskriminan – indonezijščina" lang="id" hreflang="id" data-title="Diskriminan" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonezijščina" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Diskriminanto" title="Diskriminanto – ido" lang="io" hreflang="io" data-title="Diskriminanto" data-language-autonym="Ido" data-language-local-name="ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Discriminante" title="Discriminante – italijanščina" lang="it" hreflang="it" data-title="Discriminante" data-language-autonym="Italiano" data-language-local-name="italijanščina" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%88%A4%E5%88%A5%E5%BC%8F" title="判別式 – japonščina" lang="ja" hreflang="ja" data-title="判別式" data-language-autonym="日本語" data-language-local-name="japonščina" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант – kazaščina" lang="kk" hreflang="kk" data-title="Дискриминант" data-language-autonym="Қазақша" data-language-local-name="kazaščina" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%8C%90%EB%B3%84%EC%8B%9D" title="판별식 – korejščina" lang="ko" hreflang="ko" data-title="판별식" data-language-autonym="한국어" data-language-local-name="korejščina" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Diskriminantas" title="Diskriminantas – litovščina" lang="lt" hreflang="lt" data-title="Diskriminantas" data-language-autonym="Lietuvių" data-language-local-name="litovščina" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Diskriminants" title="Diskriminants – latvijščina" lang="lv" hreflang="lv" data-title="Diskriminants" data-language-autonym="Latviešu" data-language-local-name="latvijščina" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Pembeza_layan" title="Pembeza layan – malajščina" lang="ms" hreflang="ms" data-title="Pembeza layan" data-language-autonym="Bahasa Melayu" data-language-local-name="malajščina" 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/Discriminant" title="Discriminant – nizozemščina" lang="nl" hreflang="nl" data-title="Discriminant" data-language-autonym="Nederlands" data-language-local-name="nizozemščina" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Diskriminant" title="Diskriminant – novonorveščina" lang="nn" hreflang="nn" data-title="Diskriminant" data-language-autonym="Norsk nynorsk" data-language-local-name="novonorveščina" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diskriminant" title="Diskriminant – knjižna norveščina" lang="nb" hreflang="nb" data-title="Diskriminant" data-language-autonym="Norsk bokmål" data-language-local-name="knjižna norveščina" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wyr%C3%B3%C5%BCnik_wielomianu" title="Wyróżnik wielomianu – poljščina" lang="pl" hreflang="pl" data-title="Wyróżnik wielomianu" data-language-autonym="Polski" data-language-local-name="poljščina" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант – ruščina" lang="ru" hreflang="ru" data-title="Дискриминант" data-language-autonym="Русский" data-language-local-name="ruščina" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D0%B8%D0%BD%D0%B0%D0%BD%D1%82" title="Дискриминант – jakutščina" lang="sah" hreflang="sah" data-title="Дискриминант" data-language-autonym="Саха тыла" data-language-local-name="jakutščina" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Discriminant" title="Discriminant – Simple English" lang="en-simple" hreflang="en-simple" data-title="Discriminant" 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/Diskriminant_(matematika)" title="Diskriminant (matematika) – slovaščina" lang="sk" hreflang="sk" data-title="Diskriminant (matematika)" data-language-autonym="Slovenčina" data-language-local-name="slovaščina" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Dallori" title="Dallori – albanščina" lang="sq" hreflang="sq" data-title="Dallori" data-language-autonym="Shqip" data-language-local-name="albanščina" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%A9%E0%AF%8D%E0%AE%AE%E0%AF%88%E0%AE%95%E0%AE%BE%E0%AE%9F%E0%AF%8D%E0%AE%9F%E0%AE%BF" title="தன்மைகாட்டி – tamilščina" lang="ta" hreflang="ta" data-title="தன்மைகாட்டி" data-language-autonym="தமிழ்" data-language-local-name="tamilščina" 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%94%E0%B8%B4%E0%B8%AA%E0%B8%84%E0%B8%A3%E0%B8%B4%E0%B8%A1%E0%B8%B4%E0%B9%81%E0%B8%99%E0%B8%99%E0%B8%95%E0%B9%8C" title="ดิสคริมิแนนต์ – tajščina" lang="th" hreflang="th" data-title="ดิสคริมิแนนต์" data-language-autonym="ไทย" data-language-local-name="tajščina" 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/Diskriminant" title="Diskriminant – turščina" lang="tr" hreflang="tr" data-title="Diskriminant" data-language-autonym="Türkçe" data-language-local-name="turščina" 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%94%D0%B8%D1%81%D0%BA%D1%80%D0%B8%D0%BC%D1%96%D0%BD%D0%B0%D0%BD%D1%82" title="Дискримінант – ukrajinščina" lang="uk" hreflang="uk" data-title="Дискримінант" data-language-autonym="Українська" data-language-local-name="ukrajinščina" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Diskriminant" title="Diskriminant – uzbeščina" lang="uz" hreflang="uz" data-title="Diskriminant" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="uzbeščina" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Bi%E1%BB%87t_th%E1%BB%A9c" title="Biệt thức – vietnamščina" lang="vi" hreflang="vi" data-title="Biệt thức" data-language-autonym="Tiếng Việt" data-language-local-name="vietnamščina" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%88%A4%E5%88%AB%E5%BC%8F" title="判别式 – kitajščina" lang="zh" hreflang="zh" data-title="判别式" 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class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Orodja strani"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Videz"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Videz</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">prestavi v stransko letvico</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">skrij</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Iz Wikipedije, proste enciklopedije</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="sl" dir="ltr"><p><b>Diskriminanta</b> (oznaka <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span>) je v <a href="/wiki/Algebra" title="Algebra">algebri</a> pojem, ki se uporablja v povezavi z <a href="/wiki/Polinom" title="Polinom">mnogočleniki</a>. Diskriminanta je izraz, ki daje podatke o vrstah ničel (korenov) mnogočlenika oziroma enačbe, če izenačimo mnogočlenik z 0. </p><p>Izraz izhaja iz <a href="/wiki/Latin%C5%A1%C4%8Dina" title="Latinščina">latinske</a> besede <i>discriminar</i>, kar pomeni razlikovati. Izraz je vpeljal <a href="/wiki/Angle%C5%BEi" title="Angleži">angleški</a> <a href="/wiki/Matematik" class="mw-redirect" title="Matematik">matematik</a> <a href="/wiki/James_Joseph_Sylvester" title="James Joseph Sylvester">James Joseph Sylvester</a> (1814 – 1897). </p><p>Najenostavneša in najbolj znana je diskriminanta kvadratnega mnogočlenika z obliko </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\,}"> <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> <mspace width="thinmathspace" /> </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/09c486dd3813ae1d6f0ae0e7a8c889eeff3eec70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.016ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c\,}"></span></dd></dl> <p>Diskriminanta kvadratnega mnogočlenika se izračuna z obrazcem </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 \Delta =\,b^{2}-4ac.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mspace width="thinmathspace" /> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =\,b^{2}-4ac.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecb54367d413720c11024ed13717ca44329ccae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.36ex; height:2.843ex;" alt="{\displaystyle \Delta =\,b^{2}-4ac.}"></span></dd></dl> <p>V tem primeru ima kvadratni mnogočlenik takrat, ko je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta >0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>></mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta >0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3d1feef33fbf3f878d8b6fe66fe6b793389383" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.584ex; height:2.176ex;" alt="{\displaystyle \Delta >0\,}"></span> dve realni ničli (korena) mnogočlenika. Kadar pa je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a25e7a9ed9193079c8be8b7559d33428a611be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.584ex; height:2.176ex;" alt="{\displaystyle \Delta =0\,}"></span> ima mnogočlenik samo eno realno ničlo, kadar pa je <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta <0\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo><</mo> <mn>0</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta <0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940a6e7dace7c777ee18b09fd75cb33af0cc1d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.584ex; height:2.176ex;" alt="{\displaystyle \Delta <0\,}"></span> mnogočlenik nima realnih ničel. </p><p>Diskriminanta mnogočlenika tretje stopnje (kubični mnogočlenik) z obliko </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^{3}+bx^{2}+cx+d\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</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> <mi>x</mi> <mo>+</mo> <mi>d</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ax^{3}+bx^{2}+cx+d\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a583ec775537a6f6f2db62cee8f5c39e332bd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.456ex; height:2.843ex;" alt="{\displaystyle ax^{3}+bx^{2}+cx+d\,}"></span></dd></dl> <p>je enaka </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^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <mi>a</mi> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mn>4</mn> <msup> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>d</mi> <mo>−</mo> <mn>27</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>18</mn> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeb1bbaeb49d598ce100748208856f3a71f7899a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:38.401ex; height:2.843ex;" alt="{\displaystyle \,b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd}"></span>.</dd></dl> <p>Diskriminante mnogočlenikov višjih stopenj imajo še več členov: tako ima diskriminanta mnogočlenika, ki pripada <a href="/w/index.php?title=Ena%C4%8Dba_%C4%8Detrte_stopnje&action=edit&redlink=1" class="new" title="Enačba četrte stopnje (stran ne obstaja)">enačbi četrte stopnje</a> (kvartna funkcija) 16 členov <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> , diskriminanta mnogočlenika, ki pripada <a href="/w/index.php?title=Ena%C4%8Dba_pete_stopnje&action=edit&redlink=1" class="new" title="Enačba pete stopnje (stran ne obstaja)">enačbi pete stopnje</a> (kvintna funkcija) pa 59 členov <sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>. </p><p>Če pa iščemo diskriminanto mnogočlenika, ki pripada enačbi šeste stopnje pa ima že 246 členov <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>. </p><p>Mnogočleniki imajo večkratne ničle (lahko tudi med komplesnimi števili), če in samo, če je njihova diskriminanta enaka 0. </p><p>To velja tudi, če imajo mnogočleniki koeficiente v <a href="/wiki/Obseg_(algebra)" title="Obseg (algebra)">obsegu</a>, ki ne vsebuje kompleksnih števil. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definicija">Definicija</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=1" title="Uredi razdelek: Definicija" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=1" title="Urejanje izvorne kode razdelka: Definicija"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Če izrazimo diskriminanto z ničlami mnogočlenika, jo lahko zapišemo kot </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_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30986cb6567792514bf307029bfc7596306d02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.625ex; height:5.843ex;" alt="{\displaystyle a_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}"></span></dd></dl> <p>kjer so </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790f9209748c2dca7ed7b81932c37c02af1dbc31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.448ex; height:2.009ex;" alt="{\displaystyle a_{n}}"></span> vodeči koeficienti</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1},\dots ,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},\dots ,r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b5912973d8993ca42177de2a9c63d764e278ff7" 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},\dots ,r_{n}}"></span> ničle mnogočlenika</li></ul> <p>Diskriminanta je simetrična funkcija glede na ničle, ker so koeficienti <a href="/w/index.php?title=Osnovni_simetri%C4%8Dni_mnogo%C4%8Dlenik&action=edit&redlink=1" class="new" title="Osnovni simetrični mnogočlenik (stran ne obstaja)">osnovni simetrični mnogočleniki</a> ničel. </p> <div class="mw-heading mw-heading2"><h2 id="Posplošitev"><span id="Posplo.C5.A1itev"></span>Posplošitev</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=2" title="Uredi razdelek: Posplošitev" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=2" title="Urejanje izvorne kode razdelka: Posplošitev"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pojem diskriminante lahko posplošimo še na druge <a href="/wiki/Algebrska_struktura" title="Algebrska struktura">algebrske strukture</a>. Med njimi so <a href="/wiki/Sto%C5%BEnica" title="Stožnica">stožnice</a>, <a href="/w/index.php?title=Kvadratna_forma&action=edit&redlink=1" class="new" title="Kvadratna forma (stran ne obstaja)">kvadratne forme</a> in <a href="/w/index.php?title=Algebrski_%C5%A1tevil%C4%8Dni_obseg&action=edit&redlink=1" class="new" title="Algebrski številčni obseg (stran ne obstaja)">algebrski številčni obsegi</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Diskriminanta_mnogočlenika"><span id="Diskriminanta_mnogo.C4.8Dlenika"></span>Diskriminanta mnogočlenika</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=3" title="Uredi razdelek: Diskriminanta mnogočlenika" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=3" title="Urejanje izvorne kode razdelka: Diskriminanta mnogočlenika"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Da bi našli obrazec za izračun diskriminante mnogočlenika v odvisnosti od njegovih koeficientov je najenostavneje, če uvedemo pojem <a href="/wiki/Rezultanta" title="Rezultanta">rezultante</a> (rezultanta dveh <a href="/w/index.php?title=Moni%C4%8Dni_mnogo%C4%8Dlenik&action=edit&redlink=1" class="new" title="Monični mnogočlenik (stran ne obstaja)">moničnih mnogočlenikov</a> (vodeči koeficient ima enak 1) je definirana kot produkt razlik ničel teh dveh mnogočlenikov). Tako kot je diskriminanta za samo mnogočlenik produkt kvadratov razlik med kvadrati posameznih ničel mnogočlenika, je tudi rezultanta dveh mnogočlenikov produkt razlik med njihovimi ničlami (koreni) in podobno kot diskriminanta dobi vrednost nič, če ima mnogočlenik večkratne ničle, tako tudi rezultanta dobi vrednost nič, če in samo, če imajo mnogočleniki skupno ničlo. </p><p>Ker ima mnogočlenik <span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/57622800a5830f5240d5b459f74f2f8be4dbdb0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.785ex; height:2.843ex;" alt="{\displaystyle p(x)\,}"></span> večkratne ničle, če in samo, če ima skupne ničle z odvodom <span class="mwe-math-element"><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"> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </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/a90ee253603587825b73c499255c4a81f1fb92ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.47ex; height:3.009ex;" alt="{\displaystyle p'(x)\,}"></span>, imata diskriminanta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(p)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(p)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f32394d1d3e509374a2f532243bfbe7394e490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.29ex; height:2.843ex;" alt="{\displaystyle D(p)\,}"></span> in rezultanta <span class="mwe-math-element"><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(p,p')\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(p,p')\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4313289993dcb24f22a140aca8e633583ce19a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.018ex; height:3.009ex;" alt="{\displaystyle R(p,p')\,}"></span> lastnost, da postaneta enaki 0, če in samo, če ima <span class="mwe-math-element"><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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa5f88a712eb9b03398066a0577fdcf33e02c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle p\,}"></span> večkratne ničle in imata isto stopnjo in da sta enaka do faktorja stopnje 1. </p><p>Korist od rezultante je v tem, da se lahko izračuna kot <a href="/wiki/Determinanta" title="Determinanta">determinanta</a> oziroma kot determinanta <a href="/wiki/Sylvestrova_matrika" title="Sylvestrova matrika">Sylvestrove matrike</a>, ki pa ima razsežnost <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-1)\times (2n-1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)\times (2n-1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c765b2d6624bab70a473128ddfa511a0d06339df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.966ex; height:2.843ex;" alt="{\displaystyle (2n-1)\times (2n-1)\,}"></span>. </p><p>Rezultanta <span class="mwe-math-element"><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(p,p')\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(p,p')\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4313289993dcb24f22a140aca8e633583ce19a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.018ex; height:3.009ex;" alt="{\displaystyle R(p,p')\,}"></span> splošnega mnogočlenika </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots +a_{1}x+a_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots +a_{1}x+a_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd34c6121da559ec5cc1bdd502e7b16915cb627" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:53.711ex; height:3.176ex;" alt="{\displaystyle p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots +a_{1}x+a_{0}}"></span> je enaka determinanti Sylvestrove matrike z razsežnostjo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2n-1)\times (2n-1)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2n-1)\times (2n-1)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c765b2d6624bab70a473128ddfa511a0d06339df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.966ex; height:2.843ex;" alt="{\displaystyle (2n-1)\times (2n-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 \left[{\begin{matrix}&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &\ldots &0\\&0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&\ldots \ &0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}\\&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &\ldots &0\\&0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&0&\ldots &0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}\\\end{matrix}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> <mo>…</mo> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> <mtext> </mtext> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> <mtext> </mtext> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>n</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> <mtext> </mtext> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> <mtext> </mtext> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> <mtext> </mtext> </mtd> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mi>n</mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>2</mn> <mo stretchy="false">)</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> <mtext> </mtext> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{matrix}&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &\ldots &0\\&0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&\ldots \ &0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}\\&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &\ldots &0\\&0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&0&\ldots &0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}\\\end{matrix}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4eb2117bb5bb86ebd9cd7ec664748945c4927bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -13.333ex; margin-bottom: -0.172ex; width:100.679ex; height:28.176ex;" alt="{\displaystyle \left[{\begin{matrix}&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &\ldots &0\\&0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}&0\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&\ldots \ &0&a_{n}&a_{n-1}&a_{n-2}&\ldots &a_{1}&a_{0}\\&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &\ldots &0\\&0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}&0&\ldots &0\\&\vdots \ &&&&&&&&\vdots \\&0&0&\ldots &0&na_{n}&(n-1)a_{n-1}&(n-2)a_{n-2}&\ldots \ &1a_{1}\\\end{matrix}}\right].}"></span></dd></dl> <p>Diskriminanta <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D(p)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(p)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f32394d1d3e509374a2f532243bfbe7394e490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.29ex; height:2.843ex;" alt="{\displaystyle D(p)\,}"></span> mnogočlenika <span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/57622800a5830f5240d5b459f74f2f8be4dbdb0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.785ex; height:2.843ex;" alt="{\displaystyle p(x)\,}"></span> je dana z </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 D(p)=(-1)^{{\frac {1}{2}}n(n-1)}{\frac {1}{a_{n}}}R(p,p').\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <msup> <mi>p</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(p)=(-1)^{{\frac {1}{2}}n(n-1)}{\frac {1}{a_{n}}}R(p,p').\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26e6e36f1eb1f3091c3c56de350b1f80ce899b2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:31.819ex; height:5.509ex;" alt="{\displaystyle D(p)=(-1)^{{\frac {1}{2}}n(n-1)}{\frac {1}{a_{n}}}R(p,p').\,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Zgled">Zgled</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=4" title="Uredi razdelek: Zgled" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=4" title="Urejanje izvorne kode razdelka: Zgled"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Poglejmo primer z <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>4</mn> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=4\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f51e881c68dba4d1f4f83aa800666617fa99234" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle n=4\,}"></span> za katerega je zgornja determinanta enaka </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 {\begin{vmatrix}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0&0\\&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0\\&0&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0&0\\&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0\\&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0\\&0&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\end{vmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <mn>3</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mn>2</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>1</mn> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{vmatrix}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0&0\\&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0\\&0&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0&0\\&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0\\&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0\\&0&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\end{vmatrix}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6285f21941f509e0f6584e478b115a2b962e0dd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.671ex; width:43.076ex; height:22.343ex;" alt="{\displaystyle {\begin{vmatrix}&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0&0\\&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}&0\\&0&0&a_{4}&a_{3}&a_{2}&a_{1}&a_{0}\\&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0&0\\&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0&0\\&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}&0\\&0&0&0&4a_{4}&3a_{3}&2a_{2}&1a_{1}\\\end{vmatrix}}.}"></span></dd></dl> <p>Diskriminanto mnogočlenika četrte stopnje se dobi z deljenjem te determinante z <span class="mwe-math-element"><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_{4}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{4}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/411cbffa73330c8b13167a58f4ae6ade660196df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.671ex; height:2.009ex;" alt="{\displaystyle a_{4}\,}"></span>. Če izrazimo diskriminanto s koreni je ta enaka </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_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msubsup> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e30986cb6567792514bf307029bfc7596306d02e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:18.625ex; height:5.843ex;" alt="{\displaystyle a_{n}^{2n-2}\prod _{i<j}{(r_{i}-r_{j})^{2}}}"></span></dd></dl> <p>kjer so </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}\cdots ,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> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}\cdots ,r_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728825dae759fbe38da8f9f22195cb2db6cc4199" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.901ex; height:2.009ex;" alt="{\displaystyle r_{1}\cdots ,r_{n}}"></span> kompleksne ničle (večkratnost je všteta) mnogočlenika <span class="mwe-math-element"><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> <mspace width="thinmathspace" /> </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/57622800a5830f5240d5b459f74f2f8be4dbdb0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.785ex; height:2.843ex;" alt="{\displaystyle p(x)\,}"></span></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 {\begin{matrix}p(x)&=&a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}\\&=&a_{n}(x-r_{1})(x-r_{2})\ldots (x-r_{n}).\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>p</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>…</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>=</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>…</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}p(x)&=&a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}\\&=&a_{n}(x-r_{1})(x-r_{2})\ldots (x-r_{n}).\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b87db22e96176c810751e790a0dfbac535f89c6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:45.69ex; height:6.509ex;" alt="{\displaystyle {\begin{matrix}p(x)&=&a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots +a_{1}x+a_{0}\\&=&a_{n}(x-r_{1})(x-r_{2})\ldots (x-r_{n}).\end{matrix}}}"></span></dd></dl> <p>Diskriminanto lahko definiramo za mnogočlenike nad poljubnim <a href="/wiki/Obseg_(algebra)" title="Obseg (algebra)">obsegom</a> na podoben način. </p> <div class="mw-heading mw-heading2"><h2 id="Lastnosti_ničel"><span id="Lastnosti_ni.C4.8Del"></span>Lastnosti ničel</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=5" title="Uredi razdelek: Lastnosti ničel" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=5" title="Urejanje izvorne kode razdelka: Lastnosti ničel"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Diskriminanta nudi dodatne podatke o vrsti in lastnostih ničel. Daje podatek o tem ali so ničle realne ali kompleksne ter racionalne ali so iracionalne. Dobimo tudi podatek o tem, če so ničle v obsegu nad katerim je mnogočlenik definiran ali so tudi nad razširjenim obsegom. To najlažje vidimo na kvadratnih in kubičnih mnogočlenikih. </p> <div class="mw-heading mw-heading3"><h3 id="Kvadratni_mnogočleniki"><span id="Kvadratni_mnogo.C4.8Dleniki"></span>Kvadratni mnogočleniki</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=6" title="Uredi razdelek: Kvadratni mnogočleniki" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=6" title="Urejanje izvorne kode razdelka: Kvadratni mnogočleniki"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Kvadratni mnogočleniki z realnimi koeficienti veljajo naslednja pravila za vrste ničel: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bba4bbddf69fb5d000a3d8a9daba0a36b5e720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta >0}"></span> mnogočlenik ima dve različni ničli nad realnimi števili</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc1b4ee97fd845583ecac5c0a2d151dfac284a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta <0}"></span> mnogočlenik ima dve (konjugirano kompleksni) ničli</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}"></span> mnogočlenik ima dvojno realno ničlo</li></ul> <div class="mw-heading mw-heading3"><h3 id="Kubični_mnogočlenik"><span id="Kubi.C4.8Dni_mnogo.C4.8Dlenik"></span>Kubični mnogočlenik</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=7" title="Uredi razdelek: Kubični mnogočlenik" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=7" title="Urejanje izvorne kode razdelka: Kubični mnogočlenik"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><div class="noprint relarticle mainarticle"><i>Glavni članek: <a href="/w/index.php?title=Kubi%C4%8Dni_mnogo%C4%8Dlenik&action=edit&redlink=1" class="new" title="Kubični mnogočlenik (stran ne obstaja)">Kubični mnogočlenik</a>.</i></div></dd></dl> <p>Pri kubičnem mnogočleniku pa veljajo naslednja pravila o vrsti ničel: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5bba4bbddf69fb5d000a3d8a9daba0a36b5e720" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta >0}"></span> mnogočlenik ima tri različne realne ničle</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcc1b4ee97fd845583ecac5c0a2d151dfac284a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta <0}"></span> mnogočlenik ima 1 realno in 2 <a href="/w/index.php?title=Konjugirano_kompleksni&action=edit&redlink=1" class="new" title="Konjugirano kompleksni (stran ne obstaja)">konjugirano kompleksni</a> ničli</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf057da503668fa097746562ae91517330ce5b58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =0}"></span> mnogočlenik ima 3 realne ničle, od njih sta najmanj dve enaki.</li></ul> <p>Lahko se zgodi, da ima mnogočlenik dvojno realno in še eno različno realno ničlo, v nasprotnem primeru so vse tri ničle enake. </p> <div class="mw-heading mw-heading2"><h2 id="Diskriminanta_stožnic"><span id="Diskriminanta_sto.C5.BEnic"></span>Diskriminanta stožnic</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=8" title="Uredi razdelek: Diskriminanta stožnic" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=8" title="Urejanje izvorne kode razdelka: Diskriminanta stožnic"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Sto%C5%BEnica" title="Stožnica">Stožnice</a> so definirane v ravninski geometriji z enačbo </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}+Bxy+Cy^{2}+Dx+Ey+F=0,\,}"> <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> <mi>y</mi> <mo>+</mo> <mi>C</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>D</mi> <mi>x</mi> <mo>+</mo> <mi>E</mi> <mi>y</mi> <mo>+</mo> <mi>F</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b6fc814e3046b0688f6f9ee5ec34a932ae7b6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:39.78ex; height:3.009ex;" alt="{\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0,\,}"></span></dd></dl> <p>Diskriminanta pa je enaka </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 \Delta =B^{2}-4AC\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo>=</mo> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta =B^{2}-4AC\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b54801fb2725a0b2b276f5d1041d6908a4a1ab02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.752ex; height:2.843ex;" alt="{\displaystyle \Delta =B^{2}-4AC\,}"></span></dd></dl> <ul><li>kadar je diskriminanta manjša od 0, enačba pomeni <a href="/wiki/Elipsa" title="Elipsa">elipso</a> ali <a href="/wiki/Kro%C5%BEnica" title="Krožnica">krožnico</a></li> <li>če je diskriminanta enaka 0, enačba pomeni <a href="/wiki/Parabola" title="Parabola">parabolo</a></li> <li>če je diskriminanta večja od 0, enačba pomeni <a href="/wiki/Hiperbola" title="Hiperbola">hiperbolo</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Opombe_in_sklici">Opombe in sklici</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=9" title="Uredi razdelek: Opombe in sklici" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=9" title="Urejanje izvorne kode razdelka: Opombe in sklici"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r5980307">.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 a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-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 a,.mw-parser-output .citation .cs1-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}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:#d33}.mw-parser-output .cs1-visible-error{color:#d33}.mw-parser-output .cs1-maint{display:none;color:#3a3;margin-left:0.3em}.mw-parser-output .cs1-format{font-size:95%}.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}</style><cite id="CITEREFWang2004" class="citation book cs1">Wang, Dongming (2004). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=ucpk6oO5GN0C"><i>Elimination practice: software tools and applications</i></a>. Imperial College Press. str. 180. <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:ViriKnjig/1-860-94438-8" title="Posebno:ViriKnjig/1-860-94438-8"><bdi>1-860-94438-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=Elimination+practice%3A+software+tools+and+applications&rft.pages=180&rft.pub=Imperial+College+Press&rft.date=2004&rft.isbn=1-860-94438-8&rft.aulast=Wang&rft.aufirst=Dongming&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Ducpk6oO5GN0C&rfr_id=info%3Asid%2Fsl.wikipedia.org%3ADiskriminanta" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=ucpk6oO5GN0C&pg=PA180">Chapter 10 page 180</a></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFGelfandKapranovZelevinsky1994" class="citation book cs1">Gelfand, I. M.; Kapranov, M. M.; Zelevinsky, A. V. (1994). <a rel="nofollow" class="external text" href="http://blms.oxfordjournals.org/cgi/reprint/28/1/96"><i>Discriminants, resultants and multidimensional determinants</i></a>. Birkhäuser. str. 1. <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:ViriKnjig/3-7643-3660-9" title="Posebno:ViriKnjig/3-7643-3660-9"><bdi>3-7643-3660-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=Discriminants%2C+resultants+and+multidimensional+determinants&rft.pages=1&rft.pub=Birkh%C3%A4user&rft.date=1994&rft.isbn=3-7643-3660-9&rft.aulast=Gelfand&rft.aufirst=I.+M.&rft.au=Kapranov%2C+M.+M.&rft.au=Zelevinsky%2C+A.+V.&rft_id=http%3A%2F%2Fblms.oxfordjournals.org%2Fcgi%2Freprint%2F28%2F1%2F96&rfr_id=info%3Asid%2Fsl.wikipedia.org%3ADiskriminanta" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://blms.oxfordjournals.org/cgi/pdf_extract/28/1/96">Preview page 1</a></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r5980307"><cite id="CITEREFDickensteinEmiris2005" class="citation book cs1">Dickenstein, Alicia; Emiris, Ioannis Z. (2005). <a rel="nofollow" class="external text" href="http://books.google.com/books?id=rSs-pQNrO_YC"><i>Solving polynomial equations: foundations, algorithms, and applications</i></a>. Springer. str. 26. <a href="/wiki/ISBN_(identifikator)" class="mw-redirect" title="ISBN (identifikator)">ISBN</a> <a href="/wiki/Posebno:ViriKnjig/3-540-24326-7" title="Posebno:ViriKnjig/3-540-24326-7"><bdi>3-540-24326-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=knjiga&rft.btitle=Solving+polynomial+equations%3A+foundations%2C+algorithms%2C+and+applications&rft.pages=26&rft.pub=Springer&rft.date=2005&rft.isbn=3-540-24326-7&rft.aulast=Dickenstein&rft.aufirst=Alicia&rft.au=Emiris%2C+Ioannis+Z.&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DrSs-pQNrO_YC&rfr_id=info%3Asid%2Fsl.wikipedia.org%3ADiskriminanta" class="Z3988"></span>, <a rel="nofollow" class="external text" href="http://books.google.com/books?id=rSs-pQNrO_YC&pg=PA26">Chapter 1 page 26</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Zunanje_povezave">Zunanje povezave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diskriminanta&veaction=edit&section=10" title="Uredi razdelek: Zunanje povezave" class="mw-editsection-visualeditor"><span>uredi</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Diskriminanta&action=edit&section=10" title="Urejanje izvorne kode razdelka: Zunanje povezave"><span>uredi kodo</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://www2.arnes.si/~mpavle1/mp/kvad_f.html">Diskriminanta kvadratne funkcije</a> <span class="languageicon">(slovensko)</span></li> <li><a rel="nofollow" class="external text" href="https://planetmath.org/encyclopedia/Discriminant.html">Diskriminanta na PlanethMath</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080714105224/http://planetmath.org/encyclopedia/Discriminant.html">Arhivirano</a> 2008-07-14 na <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. <span class="languageicon">(angleško)</span></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/PolynomialDiscriminant.html">Diskriminanta mnogočlenika na MathWorld</a> <span class="languageicon">(angleško)</span></li> <li><a rel="nofollow" class="external text" href="http://www.psychstat.missouristate.edu/multibook/mlt03.htm">Diskriminantna funkcijska analiza</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110604021140/http://www.psychstat.missouristate.edu/multibook/mlt03.htm">Arhivirano</a> 2011-06-04 na <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. <span class="languageicon">(angleško)</span></li> <li><a rel="nofollow" class="external text" href="http://www.numericana.com/answer/matrix.htm#discriminant">Matrike in determinante na Final Answers</a> <span class="languageicon">(angleško)</span></li></ul> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Wikipedija:Normativna_kontrola" title="Wikipedija:Normativna kontrola">Normativna kontrola: Narodne knjižnice</a> <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q192487#identifiers" title="Uredite to na Wikipodatkih"><img alt="Uredite to na Wikipodatkih" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4150190-1">Nemčija</a></span></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; 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Diskriminanta - Wikipedija, prosta enciklopedija