{"id":5855,"date":"2026-10-15T07:04:32","date_gmt":"2026-10-15T04:04:32","guid":{"rendered":"https:\/\/wpcalc.com\/fr\/mathematiques\/division-euclidienne-polynomes-2\/"},"modified":"2026-10-15T07:04:32","modified_gmt":"2026-10-15T04:04:32","slug":"division-euclidienne-polynomes-2","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/fr\/mathematiques\/division-euclidienne-polynomes-2\/","title":{"rendered":"Calculateur de division euclidienne de polyn\u00f4mes"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Diviser un polyn\u00f4me par un autre polyn\u00f4me<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"5855\" data-favorite-title=\"Division euclidienne de polyn\u00f4mes\" data-favorite-url=\"https:\/\/wpcalc.com\/fr\/mathematiques\/division-euclidienne-polynomes-2\/\" data-favorite-excerpt=\"Divisez un polyn\u00f4me par un autre, \u00e9tape par \u00e9tape. Saisissez les coefficients du dividende et du diviseur (puissance la plus \u00e9lev\u00e9e d\u2019abord, z\u00e9ros compris) et obtenez le quotient, le reste et le d\u00e9tail de chaque \u00e9tape de la division.\" aria-label=\"Ajouter aux favoris\" data-favorite-icon=\"icon icon-algebra\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/fr\/mathematiques\/division-euclidienne-polynomes-2\/\" method=\"POST\" class=\"calculator\" id=\"calculator-5855\" data-post=\"5855\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Champs de saisie<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">Coefficients du dividende (puissances d\u00e9croissantes)<\/label><textarea id=\"field-1\" name=\"coefficients_du_dividende_puissances_dcroissantes\">1; 0; -7; 6<\/textarea><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Coefficients du diviseur (puissances d\u00e9croissantes)<\/label><textarea id=\"field-2\" name=\"coefficients_du_diviseur_puissances_dcroissantes\">1; -2<\/textarea><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Variable<\/label><div class=\"grid grid-cols-1\"><select id=\"field-3\" name=\"variable\"><option value=\"x\" selected>x<\/option><option value=\"t\">t<\/option><option value=\"n\">n<\/option><option value=\"y\">y<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Recalculer\" id=\"calculate-button\" data-post=\"5855\">Calculer<\/button><button type=\"reset\">R\u00e9initialiser<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculer automatiquement<\/label><small>Si activ\u00e9, le r\u00e9sultat se met \u00e0 jour automatiquement d\u00e8s que vous modifiez une valeur.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> R\u00e9sultats de la division<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Quotient<\/span><span class=\"result-term\">Q(x)<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Reste<\/span><span class=\"result-term\">R(x)<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-3\"><span class=\"field-title\"><span>\u00c9tapes de la division<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-4\"><span class=\"field-title\"><span>V\u00e9rification\u00a0: D(x) \u00b7 Q(x) + R(x)<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copier le r\u00e9sultat\" aria-label=\"Copier le r\u00e9sultat\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copier le lien vers ce calcul<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Laisser un commentaire<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"5855\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p 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aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">D\u00e9\u00e7u <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inexact <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Formule de la division euclidienne des polyn\u00f4mes<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formule<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\nP(x) = D(x) \\cdot Q(x) + R(x)\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>O\u00f9\u00a0:<\/strong><\/p>\n<ul>\n<li>\\(P(x)\\) \u2014 le dividende, c\u2019est-\u00e0-dire le polyn\u00f4me que l\u2019on divise<\/li>\n<li>\\(D(x)\\) \u2014 le diviseur, qui ne doit pas \u00eatre le polyn\u00f4me nul<\/li>\n<li>\\(Q(x)\\) \u2014 le quotient<\/li>\n<li>\\(R(x)\\) \u2014 le reste, avec \\(\\deg R &lt; \\deg D\\)<\/li>\n<\/ul>\n<p>La division euclidienne d\u00e9termine l\u2019unique couple quotient\u2013reste v\u00e9rifiant cette identit\u00e9, de la m\u00eame fa\u00e7on que la division des entiers donne \\(a = b \\cdot q + r\\) avec \\(0 \\le r &lt; b\\). Si \\(R(x) = 0\\), le diviseur divise exactement le dividende et les racines de \\(D(x)\\) sont aussi des racines de \\(P(x)\\).<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Division euclidienne de polyn\u00f4mes \u2013 Exemple de calcul<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Exemple\u00a0:<\/strong> divisons \\(P(x) = x^3 - 7x + 6\\) par \\(D(x) = x - 2\\).<\/p>\n<p>Le terme \\(x^2\\) manquant s\u2019\u00e9crit \\(0x^2\\)\u00a0; les coefficients du dividende sont donc 1\u00a0; 0\u00a0; \u22127\u00a0; 6 et ceux du diviseur 1\u00a0; \u22122. \u00c9tape par \u00e9tape\u00a0:<\/p>\n<p>\\[x^3 \\div x = x^2, \\quad (x-2)\\,x^2 = x^3 - 2x^2, \\quad (x^3 - 7x + 6) - (x^3 - 2x^2) = 2x^2 - 7x + 6\\]<\/p>\n<p>\\[2x^2 \\div x = 2x, \\quad (x-2)\\,2x = 2x^2 - 4x, \\quad (2x^2 - 7x + 6) - (2x^2 - 4x) = -3x + 6\\]<\/p>\n<p>\\[-3x \\div x = -3, \\quad (x-2)(-3) = -3x + 6, \\quad (-3x + 6) - (-3x + 6) = 0\\]<\/p>\n<p><strong>R\u00e9sultat\u00a0:<\/strong> \\(Q(x) = x^2 + 2x - 3\\) et \\(R(x) = 0\\). La v\u00e9rification \\((x-2)(x^2+2x-3) = x^3 - 7x + 6\\) confirme que la division est exacte, donc \\(x = 2\\) est une racine de \\(P(x)\\).<\/p>\n<p>\n\r\n<\/section>\n<h2>Comment utiliser le calculateur<\/h2>\n<ol>\n<li>Saisissez les <strong>coefficients du dividende (puissance la plus \u00e9lev\u00e9e d\u2019abord)<\/strong> dans le premier champ. Mettez un 0 pour chaque puissance manquante\u00a0: \\(x^3 - 7x + 6\\) devient <code>1; 0; -7; 6<\/code>.<\/li>\n<li>Saisissez les <strong>coefficients du diviseur (puissance la plus \u00e9lev\u00e9e d\u2019abord)<\/strong> dans le second champ. Pour \\(x - 2\\), entrez <code>1; -2<\/code>\u00a0; le coefficient dominant ne doit pas \u00eatre nul.<\/li>\n<li>S\u00e9parez les nombres par des espaces, des virgules ou des points-virgules \u2014 le calculateur accepte les trois.<\/li>\n<li>Choisissez la <strong>Variable<\/strong> (x, t, n ou y) utilis\u00e9e pour afficher les polyn\u00f4mes.<\/li>\n<li>Lisez les r\u00e9sultats\u00a0: le <strong>quotient<\/strong>, le <strong>reste<\/strong>, le d\u00e9tail des <strong>\u00e9tapes de la division<\/strong> ligne par ligne, et la v\u00e9rification <strong>D(x) \u00b7 Q(x) + R(x)<\/strong>, qui doit redonner le dividende.<\/li>\n<\/ol>\n<h2>Qu\u2019est-ce que la division euclidienne de polyn\u00f4mes\u00a0?<\/h2>\n<p>La division euclidienne des polyn\u00f4mes est l\u2019analogue alg\u00e9brique de la division pos\u00e9e classique des nombres. \u00c0 partir d\u2019un dividende \\(P(x)\\) et d\u2019un diviseur \\(D(x)\\), elle fournit un quotient \\(Q(x)\\) et un reste \\(R(x)\\) tels que \\(P(x) = D(x)\\,Q(x) + R(x)\\), le reste \u00e9tant de degr\u00e9 strictement inf\u00e9rieur \u00e0 celui du diviseur. Ce couple quotient\u2013reste est unique, c\u2019est pourquoi la v\u00e9rification \\(D(x)\\,Q(x) + R(x)\\) reconstruit toujours le dividende de d\u00e9part.<\/p>\n<p>L\u2019algorithme r\u00e9p\u00e8te un cycle tr\u00e8s simple\u00a0: on divise le terme dominant courant par le terme dominant du diviseur, on multiplie tout le diviseur par ce r\u00e9sultat, puis on soustrait. Chaque cycle \u00e9limine la plus haute puissance restante du dividende et abaisse d\u2019un degr\u00e9 le reste en cours de calcul, jusqu\u2019\u00e0 ce que son degr\u00e9 passe sous celui du diviseur. La division s\u2019arr\u00eate alors, et ce qui subsiste est le reste.<\/p>\n<p>L\u2019interpr\u00e9tation du r\u00e9sultat d\u00e9pend du reste. Un reste nul signifie que \\(D(x)\\) est un facteur de \\(P(x)\\) \u2014 c\u2019est le principe des <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/regle-des-signes-de-descartes\/\">tests de factorisation et de racines<\/a> et de la division synth\u00e9tique. Un reste non nul est instructif\u00a0: d\u2019apr\u00e8s le th\u00e9or\u00e8me du reste, diviser par \\(x - c\\) laisse exactement \\(P(c)\\)\u00a0; le reste est donc la valeur du polyn\u00f4me en \\(c\\).<\/p>\n<p>La division de polyn\u00f4mes intervient partout o\u00f9 il faut les simplifier\u00a0: d\u00e9composition en \u00e9l\u00e9ments simples en calcul int\u00e9gral, recherche des asymptotes de fonctions rationnelles, r\u00e9duction d\u2019un polyn\u00f4me modulo un autre. C\u2019est aussi un moyen pratique de factoriser des polyn\u00f4mes de degr\u00e9 \u00e9lev\u00e9 d\u00e8s qu\u2019une racine est connue ou pressentie.<\/p>\n<h2>Conseils, hypoth\u00e8ses et limites<\/h2>\n<ul>\n<li><strong>Ajoutez toujours les z\u00e9ros.<\/strong> \\(x^3 - 7x + 6\\) doit \u00eatre saisi comme <code>1; 0; -7; 6<\/code>, et non <code>1; -7; 6<\/code>. Un z\u00e9ro manquant d\u00e9cale tous les coefficients vers la mauvaise puissance et donne un r\u00e9sultat faux.<\/li>\n<li><strong>Le diviseur ne doit pas \u00eatre nul.<\/strong> Son coefficient dominant ne peut pas valoir z\u00e9ro\u00a0; le calculateur supprime automatiquement les z\u00e9ros de t\u00eate, mais un diviseur enti\u00e8rement nul est refus\u00e9.<\/li>\n<li><strong>Un diviseur de degr\u00e9 trop \u00e9lev\u00e9 renvoie le dividende.<\/strong> Si \\(\\deg D &gt; \\deg P\\), le quotient vaut 0 et le reste est \u00e9gal au dividende \u2014 c\u2019est bien le bon r\u00e9sultat, pas une erreur.<\/li>\n<li><strong>Les coefficients peuvent \u00eatre n\u2019importe quels nombres r\u00e9els<\/strong>, fractions et d\u00e9cimaux compris (par ex. <code>0.5; -3; 2<\/code>). Les r\u00e9sultats sont arrondis \u00e0 4 d\u00e9cimales \u00e0 l\u2019affichage\u00a0; des d\u00e9cimales tr\u00e8s longues peuvent donc para\u00eetre l\u00e9g\u00e8rement tronqu\u00e9es.<\/li>\n<li><strong>La division par un mon\u00f4me<\/strong> comme \\(x^2\\) fonctionne aussi \u2014 saisissez <code>1; 0<\/code> comme diviseur\u00a0; chaque terme du dividende est simplement abaiss\u00e9 d\u2019un degr\u00e9.<\/li>\n<li>Pour v\u00e9rifier la multiplication de la ligne de contr\u00f4le \u00e0 la main, les calculateurs de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/multiplication-binomes\/\">multiplication de bin\u00f4mes<\/a> et de <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculateur-methode-foil\/\">m\u00e9thode FOIL<\/a> couvrent les cas les plus courants, et le <a href=\"https:\/\/wpcalc.com\/fr\/mathematiques\/calculatrice-division\/\">calculateur de division<\/a> s\u2019occupe de la division num\u00e9rique simple.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Divisez un polyn\u00f4me par un autre, \u00e9tape par \u00e9tape. Saisissez les coefficients du dividende et du diviseur (puissance la plus \u00e9lev\u00e9e d\u2019abord, z\u00e9ros compris) et obtenez le quotient, le reste et le d\u00e9tail de chaque \u00e9tape de la division.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[2],"tags":[52],"class_list":["post-5855","post","type-post","status-publish","format-standard","hentry","category-mathematiques","tag-algebre"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5855","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/comments?post=5855"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5855\/revisions"}],"predecessor-version":[{"id":5925,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/posts\/5855\/revisions\/5925"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/media?parent=5855"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/categories?post=5855"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/fr\/wp-json\/wp\/v2\/tags?post=5855"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}