{"id":27152,"date":"2021-03-28T13:00:47","date_gmt":"2021-03-28T13:00:47","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27152"},"modified":"2024-06-15T09:08:28","modified_gmt":"2024-06-15T09:08:28","slug":"algebre-des-polynomes-de-nombres-reels","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/","title":{"rendered":"Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px; text-align:center;\">\n<h1>Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els<\/h1>\n<p>    <em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\n        Dans ce cours, nous explorerons l&#8217;alg\u00e8bre des polyn\u00f4mes, sa d\u00e9finition, ses propri\u00e9t\u00e9s et ses applications. Les polyn\u00f4mes sont une partie fondamentale des math\u00e9matiques et ont de nombreuses applications dans diverses disciplines.<br \/>\n    <\/em><\/p>\n<p>    <strong>OBJECTIFS D&#8217;APPRENTISSAGE<\/strong><\/p>\n<p>\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<p style=\"text-align:left;\">\n        1. D\u00e9finir et comprendre les polyn\u00f4mes et leurs propri\u00e9t\u00e9s.<br \/>\n        2. Identifier le degr\u00e9 et les coefficients d&#8217;un polyn\u00f4me.<br \/>\n        3. R\u00e9aliser des op\u00e9rations alg\u00e9briques avec des polyn\u00f4mes et appliquer leurs propri\u00e9t\u00e9s dans des contextes math\u00e9matiques.\n    <\/p>\n<p>    <strong>TABLE DES MATI\u00c8RES :<\/strong><\/p>\n<p>\n        <a href=\"#1\"><strong>1. Alg\u00e8bre des polyn\u00f4mes : D\u00e9finitions<\/strong><\/a><br \/>\n        <a href=\"#2\"><strong>2. Types de polyn\u00f4mes<\/strong><\/a><br \/>\n        <a href=\"#3\"><strong>3. Alg\u00e8bre des polyn\u00f4mes : Op\u00e9rations<\/strong><\/a><br \/>\n        <a href=\"#4\"><strong>4. Factorisation et division des polyn\u00f4mes<\/strong><\/a>\n    <\/p>\n<p>    <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ry4sKaS3RMc\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2><strong>1. Alg\u00e8bre des polyn\u00f4mes : D\u00e9finitions<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=139s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Pour comprendre l&#8217;alg\u00e8bre des polyn\u00f4mes, nous devons d&#8217;abord savoir ce que sont les polyn\u00f4mes.<\/span><\/strong><\/a> Les polyn\u00f4mes sont des fonctions alg\u00e9briques. Si <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> est une variable r\u00e9elle, alors on dit que la fonction <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> est un polyn\u00f4me si elle peut \u00eatre \u00e9crite sous la forme :\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle P(x)= \\sum_{i=0}^n a_i x^i= a_0 + a_1x + a_2x^2 + a_3x^3 + \\cdots + a_nx^n,<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> est un entier non n\u00e9gatif et tous les <span class=\"katex-eq\" data-katex-display=\"false\">a_i<\/span>, avec <span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,2,3,\\cdots,n\\},<\/span> sont des coefficients r\u00e9els. S&#8217;il existe un <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> tel que <span class=\"katex-eq\" data-katex-display=\"false\">a_k\\neq 0<\/span> et, lorsque <span class=\"katex-eq\" data-katex-display=\"false\">k\\lt i<\/span>, il arrive que <span class=\"katex-eq\" data-katex-display=\"false\">a_i=0<\/span>, alors on dit que cette valeur de <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> est <strong>le degr\u00e9 du polyn\u00f4me.<\/strong> En d&#8217;autres termes, le degr\u00e9 d&#8217;un polyn\u00f4me est la plus grande puissance qui accompagne un coefficient non nul.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><strong>2. Types de polyn\u00f4mes<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=340s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Les polyn\u00f4mes sont class\u00e9s selon leur degr\u00e9 ;<\/span><\/strong><\/a> c&#8217;est pourquoi, lorsqu&#8217;un polyn\u00f4me est mentionn\u00e9, on dit presque toujours qu&#8217;il s&#8217;agit d&#8217;un polyn\u00f4me de degr\u00e9 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span>, lorsque <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> est la plus grande puissance de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> qui accompagne le coefficient non nul de ce polyn\u00f4me.\n<\/p>\n<h3>2.1. Les polyn\u00f4mes constants<\/h3>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est la famille qui englobe tous les polyn\u00f4mes de degr\u00e9 z\u00e9ro et le polyn\u00f4me nul. On dit qu&#8217;un polyn\u00f4me est de degr\u00e9 z\u00e9ro, s&#8217;il peut \u00eatre \u00e9crit sous la forme <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=c,<\/span> avec <span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0.<\/span> D&#8217;autre part, le polyn\u00f4me nul est de la forme <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 0<\/span> et pour celui-ci, un degr\u00e9 n&#8217;est pas d\u00e9fini.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><strong>3. Alg\u00e8bre des polyn\u00f4mes : Op\u00e9rations<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=428s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Les polyn\u00f4mes h\u00e9ritent toutes leurs propri\u00e9t\u00e9s de l&#8217;alg\u00e8bre des nombres r\u00e9els.<\/span><\/strong><\/a> Les propri\u00e9t\u00e9s distributives et associatives sont particuli\u00e8rement pertinentes.\n<\/p>\n<h3>3.1. Addition et soustraction<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=470s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> sont deux polyn\u00f4mes de degr\u00e9<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span>, respectivement, avec<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">m=n+k<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">0\\leq k,<\/span>\n<p style=\"text-align: justify; color: #000000;\">alors il se trouve que :<\/p>\n<p style=\"text-align: center; color: #000000;\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle P(x) \\pm Q(x) &amp;=\\displaystyle \\sum_{i=0}^n a_i x^i \\pm \\sum_{i=0}^m b_i x^i \\\\ \\\\\n\n &amp;\\displaystyle = \\sum_{i=0}^n a_i x^i \\pm \\left( \\sum_{i=0}^n b_i x^i + \\sum_{i=n+1}^{n+k} b_i x^i \\right) \\\\ \\\\\n\n&amp;\\displaystyle = \\sum_{i=0}^n (a_i \\pm b_i) x^i + \\sum_{i=n+1}^m b_i x^i\n\n\\end{array}\n\n<\/span>\n<p style=\"text-align: justify; color: #000000;\">C&#8217;est-\u00e0-dire que les coefficients accompagnant les m\u00eames puissances de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> sont additionn\u00e9s ou soustraits, selon le cas.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">EXEMPLE :<\/span><br \/>\nSi <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3+5x+2x^2<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 6x-3x^2 +23x^5<\/span>, alors :<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) + Q(x) = \\cdots \\\\ = (3+5x+2x^2) + (6x-3x^2 +23x^5) \\\\ = 3 + (5+6)x + (2-3)x^2 + 23x^5 \\\\ = 3 + 11x - x^2 + 23x^5 <\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x) - Q(x) = \\cdots \\\\ = (3+5x+2x^2) - (6x-3x^2 +23x^5) \\\\ = 3 + (5-6)x + (2+3)x^2 - 23x^5 \\\\ = 3 - x + 5x^2 - 23x^5 <\/span>\n<\/p>\n<h3>3.2. Multiplication<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=894s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Dans le m\u00eame contexte que pour l&#8217;addition et la soustraction de polyn\u00f4mes,<\/span><\/strong><\/a> la multiplication des polyn\u00f4mes se d\u00e9veloppera de la mani\u00e8re suivante :\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Nous distinguons d&#8217;abord la multiplication par un scalaire. Si <span class=\"katex-eq\" data-katex-display=\"false\">c \\in \\mathbb{R},<\/span> alors nous avons :\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle c P(x) = c \\sum_{i=0}^n a_i x^i =\\sum_{i=0}^n c a_i x^i <\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Et puis nous avons la multiplication entre polyn\u00f4mes :\n<\/p>\n<p style=\"text-align: center; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\displaystyle P(x) Q(x) &amp;\\displaystyle = \\left( \\sum_{i=0}^n a_i x^i \\right) \\left(\\sum_{j=0}^m b_j x^j\\right) \\\\ \\\\\n\n&amp;=\\displaystyle \\left[\\sum_{j=0}^m \\left( \\ sum_{i=0}^n a_i x^i \\right) b_j x^j\\right] \\\\ \\\\\n\n&amp;=\\displaystyle \\ sum_{j=0}^m \\left( \\ sum_{i=0}^n a_ib_j x^{i+j} \\ right) \\\\ \\\\\n\n&amp;=\\displaystyle \\ sum_{i,j=0}^{n,m} a_ib_j x^{i+j}\n\n\\end{array}<\/span>\n<p style=\"text-align: justify; color: #000000;\">\n    C&#8217;est ce que nous r\u00e9sumerions par l&#8217;expression \u00abla somme des produits de tous avec tous\u00bb.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXEMPLE :<\/span><br \/>\n    Si <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 4x+ 2x^2-x^4<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 5 - x + x^2-7x^3,<\/span> alors :\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">P(x)Q(x) =\\cdots \\\\ {} \\\\= (4x+ 2x^2-x^4)(5 - x + x^2-7x^3) \\\\ {} \\\\ = 4x(5 - x + x^2-7x^3) \\\\ + 2x^2 (5 - x + x^2-7x^3) \\\\ - x^4 (5 - x + x^2-7x^3) \\\\ {} \\\\ = 20x - 4x^2 + 4x^3 - 28x^4 \\\\ + 10x^2 - 2x^3 + 2x^4 - 14x^5 \\\\ -5x^4 + x^5 - x^6 + 7x^7 \\\\ {} \\\\ = 20x + 6x^2 + 2x^3 - 31x^4 - 13x^5 - x^6 + 7x^7<\/span>\n<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2><strong>4. Factorisation et division des polyn\u00f4mes<\/strong><\/h2>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1375s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Lorsque nous multiplions deux polyn\u00f4mes, nous passons de deux polyn\u00f4mes simples \u00e0 un autre plus complexe (de degr\u00e9 sup\u00e9rieur).<\/span><\/strong><\/a> Lorsque nous factorisons un polyn\u00f4me, nous suivons le processus inverse : nous transformons un polyn\u00f4me complexe en produit de deux ou plusieurs polyn\u00f4mes de degr\u00e9 inf\u00e9rieur.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Pour factoriser un polyn\u00f4me <span class=\"katex-eq\" data-katex-display=\"false\">P(x),<\/span> il est n\u00e9cessaire de trouver les valeurs de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> qui annulent le polyn\u00f4me ; si de telles valeurs existent, alors le polyn\u00f4me est factorisable. Parler de l&#8217;existence est accessible, mais les trouver est une autre histoire. Nous examinerons ce sujet plus en d\u00e9tail lorsque nous \u00e9tudierons les factorisations des polyn\u00f4mes quadratiques et (2n)quadratiques.\n<\/p>\n<h3>4.1. Produits notables<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1654s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">Cependant, il existe des cas o\u00f9 la factorisation est obtenue facilement,<\/span><\/strong><br \/>\n    <\/a> comme celui des produits notables. Certains de ces r\u00e9sultats sont les suivants :\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^2 - y^2 = (x-y)(x+y)<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x\\pm y)^2 = x^2 \\pm 2xy + y^2<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">(x \\pm y)^3 = x^3 \\pm 3x^2y + 3xy^2 \\pm y^3<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3-y^3=(x-y)(x^2+xy+y^2)<\/span>\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span class=\"katex-eq\" data-katex-display=\"false\">x^3+y^3=(x+y)(x^2-xy+y^2)<\/span>\n<\/p>\n<h3>4. L&#8217;algorithme de la division<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\n    <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1854s\" target=\"_blank\" rel=\"noopener\"><br \/>\n        <strong><span style=\"color: #ff0000;\">De m\u00eame que la multiplication d&#8217;entiers donne des nombres compos\u00e9s et que la division via l&#8217;algorithme de la division permet de factoriser lorsque le reste est nul,<\/span><\/strong><\/a> quelque chose de similaire se produit avec les polyn\u00f4mes. Expliquer l&#8217;algorithme de la division \u00aben texte\u00bb peut \u00eatre un peu compliqu\u00e9, il est beaucoup plus facile de le comprendre en regardant directement comment cela se fait et dans quels cas l&#8217;algorithme conduit \u00e0 une factorisation. Pour cela, nous examinerons quelques exemples.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXEMPLE :<\/span> Calculer <span class=\"katex-eq\" data-katex-display=\"false\">P(x):Q(x)<\/span> pour les cas suivants :\n<\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=2 x^3 + x^2 - 2 x - 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x-1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=1930s\" target=\"_blank\" rel=\"noopener\"> <strong><span style=\"color: #ff0000;\">[SOLUTION]<\/span><\/strong> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^4+2x^3-x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2-4<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2120s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=3 x^4 - 2 x^3 - x^2 - 4 x + 1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^2+x+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2331s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<li>\n        <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=x^7+5x^4+5x^2-3x+1, <\/span> <span class=\"katex-eq\" data-katex-display=\"false\">Q(x)=x^3-2x^2+1<\/span> <a href=\"https:\/\/www.youtube.com\/watch?v=ry4sKaS3RMc&amp;t=2464s\" target=\"_blank\" rel=\"noopener\"> <span style=\"color: #ff0000;\"><strong>[SOLUTION]<\/strong><\/span> <\/a>\n    <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els R\u00e9sum\u00e9 : Dans ce cours, nous explorerons l&#8217;alg\u00e8bre des polyn\u00f4mes, sa d\u00e9finition, ses propri\u00e9t\u00e9s et ses applications. Les polyn\u00f4mes sont une partie fondamentale des math\u00e9matiques et ont de nombreuses applications dans diverses disciplines. OBJECTIFS D&#8217;APPRENTISSAGE \u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de : 1. D\u00e9finir et [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27134,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":2,"footnotes":""},"categories":[585,569],"tags":[],"class_list":["post-27152","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebre-et-geometrie","category-mathematiques"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez l&#039;alg\u00e8bre des polyn\u00f4mes : d\u00e9finitions, types, op\u00e9rations, produits, factorisation et division.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els\" \/>\n<meta property=\"og:description\" content=\"Apprenez l&#039;alg\u00e8bre des polyn\u00f4mes : d\u00e9finitions, types, op\u00e9rations, produits, factorisation et division.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-03-28T13:00:47+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-15T09:08:28+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els\" \/>\n<meta name=\"twitter:description\" content=\"Apprenez l&#039;alg\u00e8bre des polyn\u00f4mes : d\u00e9finitions, types, op\u00e9rations, produits, factorisation et division.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els\",\"datePublished\":\"2021-03-28T13:00:47+00:00\",\"dateModified\":\"2024-06-15T09:08:28+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\"},\"wordCount\":1320,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\",\"articleSection\":[\"Alg\u00e8bre et G\u00e9om\u00e9trie\",\"Math\u00e9matiques\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\",\"url\":\"http:\/\/toposuranos.com\/material\/fr\/algebre-des-polynomes-de-nombres-reels\/\",\"name\":\"Alg\u00e8bre des polyn\u00f4mes de nombres r\u00e9els - 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