{"id":28843,"date":"2021-03-30T13:00:34","date_gmt":"2021-03-30T13:00:34","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28843"},"modified":"2024-09-22T02:06:25","modified_gmt":"2024-09-22T02:06:25","slug":"factorisation-du-polynome-quadratique-et-2n-quadratique","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/fr\/factorisation-du-polynome-quadratique-et-2n-quadratique\/","title":{"rendered":"Factorisation du polyn\u00f4me quadratique et 2n-quadratique"},"content":{"rendered":"<p><center><\/p>\n<h1>Factorisation du polyn\u00f4me quadratique et 2n-quadratique<\/h1>\n<p><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\n   Dans ce cours, nous examinerons en d\u00e9tail le processus de factorisation des polyn\u00f4mes quadratiques <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span> et des polyn\u00f4mes <span class=\"katex-eq\" data-katex-display=\"false\">(2n)<\/span>-quadratiques <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>, en les d\u00e9composant en facteurs simples. Les proc\u00e9dures seront d\u00e9velopp\u00e9es math\u00e9matiquement et des exemples pratiques seront montr\u00e9s.<\/em><\/p>\n<p>   <strong>Objectifs d&#8217;apprentissage<\/strong><\/p>\n<ol style=\"text-align: left;\">\n<li><strong>Apprendre<\/strong> \u00e0 factoriser des polyn\u00f4mes quadratiques de la forme <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span>.<\/li>\n<li><strong>D\u00e9river<\/strong> et utiliser la formule quadratique <span class=\"katex-eq\" data-katex-display=\"false\">x = \\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}<\/span> pour trouver les racines.<\/li>\n<li><strong>Appliquer<\/strong> des techniques de factorisation aux polyn\u00f4mes (2n)-quadratiques de la forme <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>.<\/li>\n<li><strong>Reconna\u00eetre<\/strong> les conditions n\u00e9cessaires \u00e0 la factorisation des polyn\u00f4mes quadratiques.<\/li>\n<li><strong>Utiliser<\/strong> la m\u00e9thode de compl\u00e9tion du carr\u00e9 dans le processus de factorisation.<\/li>\n<\/li>\n<\/ol>\n<p>   <strong>TABLE DES MATI\u00c8RES :<\/strong><br \/>\n   <a href=\"#1\">Introduction<\/a><br \/>\n   <a href=\"#2\">Polyn\u00f4me quadratique et polyn\u00f4me (2n)-quadratique<\/a><br \/>\n   <a href=\"#3\">Factorisation du polyn\u00f4me quadratique<\/a><br \/>\n   <a href=\"#4\">Extension \u00e0 la factorisation du polyn\u00f4me bi-quadratique<\/a><br \/>\n   <a href=\"#5\">Exercices pratiques<\/a>\n   <\/p>\n<p><\/center><\/p>\n<p><center><br \/>\n   <iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ddTfUR7QBfY\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify;\">Apprendre \u00e0 factoriser un polyn\u00f4me quadratique est la premi\u00e8re \u00e9tape pour commencer \u00e0 \u00e9tudier de nombreuses autres techniques de factorisation. C&#8217;est pourquoi nous examinerons en profondeur cette technique et \u00e9tendrons son utilisation aussi loin que possible. \u00c0 la fin, vous aurez appris non seulement \u00e0 factoriser un polyn\u00f4me quadratique (de degr\u00e9 2), mais aussi \u00e0 utiliser ces m\u00eames techniques pour factoriser tout polyn\u00f4me (2n)-quadratique.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Polyn\u00f4me quadratique et polyn\u00f4me (2n)-quadratique<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=96s\" target=\"_blank\" rel=\"noopener\"><strong>Un polyn\u00f4me quadratique est un polyn\u00f4me de degr\u00e9 deux.<\/strong><\/a> Ainsi, un polyn\u00f4me quadratique est toute fonction de la forme<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2}+bx +c <\/span>\n<p style=\"text-align: justify;\">avec <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>. Notre \u00e9tude ne se concentrera cependant pas uniquement sur la factorisation des polyn\u00f4mes de cette forme, mais nous viserons une forme g\u00e9n\u00e9ralis\u00e9e dont le quadratique n&#8217;est qu&#8217;un cas particulier. Nous parlons du polyn\u00f4me (2n)-quadratique. Cette g\u00e9n\u00e9ralisation englobe tous les polyn\u00f4mes qui peuvent \u00eatre \u00e9crits sous la forme<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n}+bx^n +c <\/span>\n<p style=\"text-align: justify;\">o\u00f9, en plus de supposer <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> et <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>, on prend un <span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{N}<\/span> quelconque. Des exemples de ce type de polyn\u00f4me incluent :<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3x^2 -x + 1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 7x^4 +5x^2 + 3<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = -4x^6 +12x^3 + 2<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = 21x^8 -75 x^4 -9<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">et ainsi de suite.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Factorisation du polyn\u00f4me quadratique<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=193s\" target=\"_blank\" rel=\"noopener\"><strong>Comme nous l&#8217;avons d\u00e9j\u00e0 vu, un polyn\u00f4me de degr\u00e9 2 a la forme g\u00e9n\u00e9rale<\/strong><\/a><\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2}+bx +c \\;\\; , \\;\\; a\\neq 0 <\/span>\n<p style=\"text-align: justify;\">La factorisation est le processus qui consiste \u00e0 d\u00e9composer un polyn\u00f4me complexe en produit de deux polyn\u00f4mes plus simples. Ainsi, s&#8217;il est possible de factoriser, il existe des constantes <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta,\\gamma,\\delta \\in\\mathbb{R}<\/span>, avec <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\gamma \\neq 0<\/span>, telles que :<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\">= (\\alpha x + \\beta)(\\gamma x + \\delta) <\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\">= \\alpha \\gamma \\left(x +\\displaystyle \\frac{\\beta}{\\alpha}\\right)\\left(x + \\frac{\\delta}{\\gamma}\\right) <\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">Comme il y a une \u00e9galit\u00e9 entre le c\u00f4t\u00e9 gauche et le c\u00f4t\u00e9 droit, lorsque l&#8217;un des c\u00f4t\u00e9s est annul\u00e9, l&#8217;autre l&#8217;est \u00e9galement. Le c\u00f4t\u00e9 droit s&#8217;annule lorsque <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\beta\/\\alpha<\/span> ou <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\delta\/\\gamma<\/span>. Voyons maintenant pour quelles valeurs le c\u00f4t\u00e9 gauche de cette \u00e9galit\u00e9 s&#8217;annule. Nous aurons :<\/p>\n<table style=\"text-align: justify;\">\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx + c<\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = 0<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">ax^2 + bx <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = -c<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + \\displaystyle \\frac{b}{a}x <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = - \\displaystyle \\frac{c}{a}<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right; background-color: #ffc0c0;\"><span class=\"katex-eq\" data-katex-display=\"false\">x^2 + \\displaystyle \\frac{b}{a}x + \\frac{b^2}{4a^2}<\/span><\/td>\n<td style=\"text-align: left; background-color: #ffc0c0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> =\\displaystyle \\frac{b^2}{4a^2} -\\frac{c}{a} = \\frac{ab^2 - 4a^2 c}{4a^3} = \\frac{b^2 - 4ac }{4a^2}<\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\left(x + \\displaystyle \\frac{b}{2a}\\right)^2<\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\displaystyle \\frac{b^2 - 4ac }{4a^2} <\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x + \\displaystyle \\frac{b}{2a} <\/span><\/td>\n<td style=\"text-align: left;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\pm \\sqrt{\\displaystyle \\frac{b^2 - 4ac }{4a^2}} = \\frac{\\pm\\sqrt{b^2 - 4ac }}{2a} <\/span><\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: right; background-color: #a0ffa0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> x <\/span><\/td>\n<td style=\"text-align: left; background-color: #a0ffa0;\"><span class=\"katex-eq\" data-katex-display=\"false\"> = \\displaystyle \\frac{-b \\pm\\sqrt{b^2 - 4ac }}{2a} <\/span> \u2705<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p style=\"text-align: justify;\">En nous basant sur ce raisonnement, les constantes de factorisation devront satisfaire les conditions suivantes (sans perte de g\u00e9n\u00e9ralit\u00e9) :<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha\\gamma = a<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\beta}{\\alpha} = - \\left(\\frac{-b + \\sqrt{b^2 - 4ac }}{2a} \\right)<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{\\delta}{\\gamma} = - \\left(\\frac{-b - \\sqrt{b^2 - 4ac }}{2a} \\right)<\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Nous avons ainsi une technique qui permet de factoriser tout polyn\u00f4me de degr\u00e9 2. Si ce n&#8217;est pas possible, un signe n\u00e9gatif sous la racine (le discriminant) l&#8217;indiquera. Nous pouvons r\u00e9sumer cela en utilisant la notation suivante :<\/p>\n<ul style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">x_1 =\\displaystyle \\frac{-b + \\sqrt{b^2 - 4ac }}{2a} <\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">x_2 =\\displaystyle \\frac{-b - \\sqrt{b^2 - 4ac }}{2a} <\/span><\/li>\n<\/ul>\n<p style=\"text-align: justify;\">Ce qui se r\u00e9sume \u00e0 la bonne vieille formule fiable :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{x_{1,2} = \\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}}<\/span> \u2705<\/p>\n<p style=\"text-align: justify;\">Ainsi, la factorisation finale est :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{P(x) = ax^2 +bx + c = a(x-x_1)(x - x_2)}<\/span>\u2705<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Extension \u00e0 la factorisation du polyn\u00f4me bi-quadratique<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=997s\" target=\"_blank\" rel=\"noopener\"><strong>Cette technique peut \u00e9galement \u00eatre utilis\u00e9e pour factoriser un polyn\u00f4me bi-quadratique.<\/strong><\/a> Voici la m\u00e9thode :<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = ax^4 + bx^2 + c = a(x^2)^2 + bx^2 + c =a (x^2 - x_1^2)(x^2-x_2^2) <\/span>\n<p style=\"text-align: justify;\">O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\"> x^2_{1,2} = \\displaystyle \\dfrac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. On peut donc \u00e9crire :<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = ax^4 + bx^2 + c = a\\left(x^2 - \\displaystyle \\dfrac{-b + \\sqrt{b^2 - 4ac }}{2a}\\right) \\left(x^2- \\dfrac{-b - \\sqrt{b^2 - 4ac }}{2a}\\right) <\/span>\n<p style=\"text-align: justify;\">Si <span class=\"katex-eq\" data-katex-display=\"false\">x_1^2<\/span> est un nombre positif, on peut utiliser la d\u00e9composition somme-produit pour s\u00e9parer <span class=\"katex-eq\" data-katex-display=\"false\">(x^2 - x_1^2) = (x-x_1)(x + x_1)<\/span> ; sinon, on aura affaire \u00e0 des nombres complexes, et la factorisation dans les nombres r\u00e9els ne sera plus possible. Si toutes les racines sont bien d\u00e9finies, on peut \u00e9crire :<\/p>\n<p style=\"text-align: justify;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\nQ(x) &amp;= ax^4 + bx^2 + c \\\\ \\\\\n\n     &amp; = a \\left(x -\\displaystyle \\sqrt{\\frac{-b + \\sqrt{b^2 - 4ac }}{2a}}\\right) \\left(x + \\displaystyle \\sqrt{\\frac{-b + \\sqrt{b^2 - 4ac }}{2a}}\\right) \\\\ \\\\\n\n&amp; \\left(x- \\displaystyle \\sqrt{\\frac{-b - \\sqrt{b^2 - 4ac }}{2a}}\\right) \\left(x+ \\sqrt{\\displaystyle \\frac{-b - \\sqrt{b^2 - 4ac }}{2a}}\\right)\n\n\\end{array}<\/span>\n<p style=\"text-align: justify;\">Sinon, vous devrez vous arr\u00eater \u00e0 l&#8217;\u00e9tape pr\u00e9c\u00e9dente.<\/p>\n<h3>G\u00e9n\u00e9ralisation \u00e0 la factorisation du polyn\u00f4me (2n)-quadratique<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong>Avec cette technique, vous pouvez \u00e9galement factoriser un polyn\u00f4me (2n)-quadratique.<\/strong><\/a> Il suffit de r\u00e9\u00e9crire la forme et d&#8217;utiliser les m\u00e9thodes pr\u00e9c\u00e9dentes lorsque les racines sont bien d\u00e9finies. Ainsi, nous aurons :<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = a(x^n)^{2}+b (x^n) +c = a(x^n-x_1^n)(x^n-x_2^n) <\/span>\n<p style=\"text-align: justify;\">O\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">x^n_{1,2} =\\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. Ensuite, s\u00e9parez avec somme-produit l\u00e0 o\u00f9 il n&#8217;y a pas de nombres complexes.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Exercices pratiques :<\/h2>\n<p style=\"text-align: justify;\">Maintenant, c&#8217;est \u00e0 vous d&#8217;essayer ces techniques avec quelques exercices. Les polyn\u00f4mes ci-dessous ont \u00e9t\u00e9 choisis au hasard pour que vous puissiez reconna\u00eetre les difficult\u00e9s potentielles de la factorisation.<\/p>\n<h3>Premier Round<\/h3>\n<p style=\"text-align: justify;\">Ces polyn\u00f4mes sont ceux que j&#8217;ai donn\u00e9s en exemple au d\u00e9but du post :<\/p>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3x^2 -x + 1<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 7x^4 +5x^2 + 3<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = -4x^6 +12x^3 + 2<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = 21x^8 -75 x^4 -9<\/span><\/li>\n<\/ol>\n<h3>Deuxi\u00e8me Round<\/h3>\n<p style=\"text-align: justify;\">Voici quelques autres exemples un peu plus difficiles :<\/p>\n<ol style=\"text-align: justify;\">\n<li><span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 78x^2 -21x - 13<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 27x^4 +5x^2 - 14<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">R(x) = 9x^6 +12x^3 - 16<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">S(x) = -9x^8 -2 x^4 + 10<\/span><\/li>\n<li><span class=\"katex-eq\" data-katex-display=\"false\">T(x) = 5x^{12} -2 x^6 - 15<\/span><\/li>\n<\/ol>\n<h3>Solution des exercices<\/h3>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ilNTFyF7Hmo\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Factorisation du polyn\u00f4me quadratique et 2n-quadratique R\u00e9sum\u00e9 : Dans ce cours, nous examinerons en d\u00e9tail le processus de factorisation des polyn\u00f4mes quadratiques et des polyn\u00f4mes -quadratiques , en les d\u00e9composant en facteurs simples. Les proc\u00e9dures seront d\u00e9velopp\u00e9es math\u00e9matiquement et des exemples pratiques seront montr\u00e9s. Objectifs d&#8217;apprentissage Apprendre \u00e0 factoriser des polyn\u00f4mes quadratiques de la forme [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":28831,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":1,"footnotes":""},"categories":[585,569],"tags":[],"class_list":["post-28843","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>Factorisation du polyn\u00f4me quadratique et 2n-quadratique - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez comment r\u00e9aliser facilement la factorisation du polyn\u00f4me quadratique. 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