{"id":28834,"date":"2021-03-30T13:00:36","date_gmt":"2021-03-30T13:00:36","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=28834"},"modified":"2024-09-22T02:04:36","modified_gmt":"2024-09-22T02:04:36","slug":"fatoracao-do-polinomio-quadratico-e-2n-quadratico","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/fatoracao-do-polinomio-quadratico-e-2n-quadratico\/","title":{"rendered":"Fatora\u00e7\u00e3o do Polin\u00f4mio Quadr\u00e1tico e 2n-Quadr\u00e1tico"},"content":{"rendered":"<p><center><\/p>\n<h1>Fatora\u00e7\u00e3o do Polin\u00f4mio Quadr\u00e1tico e 2n-Quadr\u00e1tico<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\n   Nesta aula, revisaremos detalhadamente o processo de fatora\u00e7\u00e3o de polin\u00f4mios quadr\u00e1ticos <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span> e polin\u00f4mios <span class=\"katex-eq\" data-katex-display=\"false\">(2n)<\/span>-quadr\u00e1ticos <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>, decompondo-os em fatores simples. Ser\u00e3o desenvolvidos procedimentos matem\u00e1ticos e exemplos pr\u00e1ticos ser\u00e3o mostrados.<\/em><\/p>\n<p>   <strong>Objetivos de aprendizagem<\/strong><\/p>\n<ol style=\"text-align: left;\">\n<li><strong>Aprender<\/strong> a fatorar polin\u00f4mios quadr\u00e1ticos da forma <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^2 + bx + c<\/span>.<\/li>\n<li><strong>Derivar<\/strong> e utilizar a f\u00f3rmula quadr\u00e1tica <span class=\"katex-eq\" data-katex-display=\"false\">x = \\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}<\/span> para encontrar as ra\u00edzes.<\/li>\n<li><strong>Aplicar<\/strong> t\u00e9cnicas de fatora\u00e7\u00e3o a polin\u00f4mios (2n)-quadr\u00e1ticos da forma <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = ax^{2n} + bx^n + c<\/span>.<\/li>\n<li><strong>Reconhecer<\/strong> as condi\u00e7\u00f5es necess\u00e1rias para a fatora\u00e7\u00e3o de polin\u00f4mios de tipo quadr\u00e1tico.<\/li>\n<li><strong>Utilizar<\/strong> o m\u00e9todo de completar o quadrado no processo de fatora\u00e7\u00e3o.<\/li>\n<\/ol>\n<p>   <strong>\u00cdNDICE DE CONTE\u00daDOS:<\/strong><br \/>\n   <a href=\"#1\">Introdu\u00e7\u00e3o<\/a><br \/>\n   <a href=\"#2\">Polin\u00f4mio Quadr\u00e1tico e Polin\u00f4mio (2n)-quadr\u00e1tico<\/a><br \/>\n   <a href=\"#3\">Fatora\u00e7\u00e3o do polin\u00f4mio quadr\u00e1tico<\/a><br \/>\n   <a href=\"#4\">Expans\u00e3o para a fatora\u00e7\u00e3o do polin\u00f4mio bi-quadr\u00e1tico<\/a><br \/>\n   <a href=\"#5\">Exerc\u00edcios de exemplo<\/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><\/center><br \/>\n   <a name=\"1\"><\/a><\/p>\n<h2>Introdu\u00e7\u00e3o<\/h2>\n<p style=\"text-align: justify;\">Aprender a fatorar o polin\u00f4mio quadr\u00e1tico \u00e9 o primeiro passo para come\u00e7ar a estudar muitas outras t\u00e9cnicas de fatora\u00e7\u00e3o. Por isso, revisaremos profundamente esta t\u00e9cnica e expandiremos seu uso tanto quanto poss\u00edvel. Ao terminar, voc\u00ea ter\u00e1 aprendido n\u00e3o s\u00f3 a fatorar o polin\u00f4mio quadr\u00e1tico (de grau 2), mas tamb\u00e9m usar\u00e1 essas mesmas t\u00e9cnicas para fatorar qualquer polin\u00f4mio (2n)-quadr\u00e1tico.<\/p>\n<p>   <a name=\"2\"><\/a><\/p>\n<h2>Polin\u00f4mio Quadr\u00e1tico e Polin\u00f4mio (2n)-quadr\u00e1tico<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=96s\" target=\"_blank\" rel=\"noopener\"><strong>Um polin\u00f4mio quadr\u00e1tico \u00e9 o polin\u00f4mio de grau dois.<\/strong><\/a> A partir disso, temos que um polin\u00f4mio quadr\u00e1tico \u00e9 qualquer fun\u00e7\u00e3o da forma<\/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;\">com <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>. Nosso estudo, no entanto, n\u00e3o se concentrar\u00e1 apenas em fatorar os polin\u00f4mios dessa forma, mas apontaremos para uma forma generalizada da qual o quadr\u00e1tico \u00e9 apenas um caso particular. Estamos falando do polin\u00f4mio (2n)-quadr\u00e1tico. Esta generaliza\u00e7\u00e3o engloba todos os polin\u00f4mios que podem ser escritos na forma<\/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;\">onde, al\u00e9m de assumir <span class=\"katex-eq\" data-katex-display=\"false\">a,b,c\\in\\mathbb{R}<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">a\\neq 0<\/span>, \u00e9 tomado um <span class=\"katex-eq\" data-katex-display=\"false\">n\\in\\mathbb{N}<\/span> qualquer. Exemplos deste tipo de polin\u00f4mio seriam:<\/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;\">e assim por diante.<\/p>\n<p>   <a name=\"3\"><\/a><\/p>\n<h2>Fatora\u00e7\u00e3o do polin\u00f4mio quadr\u00e1tico<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=193s\" target=\"_blank\" rel=\"noopener\"><strong>Como j\u00e1 vimos, um polin\u00f4mio de grau 2 tem a forma geral<\/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;\">A fatora\u00e7\u00e3o \u00e9 o processo que separa um polin\u00f4mio complexo no produto de dois polin\u00f4mios mais simples. Por isso, se for poss\u00edvel fatorar, ent\u00e3o existem constantes <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha,\\beta,\\gamma,\\delta \\in\\mathbb{R}<\/span>, com <span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\gamma \\neq 0<\/span>, tais 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;\">Como temos uma igualdade entre o lado esquerdo e o direito, ent\u00e3o teremos que, quando um lado for nulo, o outro tamb\u00e9m ser\u00e1 anulado. Acontece que o lado direito \u00e9 anulado quando <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\beta\/\\alpha<\/span> ou quando <span class=\"katex-eq\" data-katex-display=\"false\">x=-\\delta\/\\gamma<\/span>. Vamos agora ver para quais valores o lado esquerdo desta igualdade \u00e9 anulado. Teremos 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\">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;\">A partir desse racioc\u00ednio, teremos que as constantes em letras gregas da fatora\u00e7\u00e3o devem satisfazer (sem perda de generalidade) as seguintes condi\u00e7\u00f5es:<\/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;\">E com isso j\u00e1 temos uma t\u00e9cnica que nos permitir\u00e1 fatorar qualquer polin\u00f4mio de grau 2, e se n\u00e3o for poss\u00edvel fatorar, ent\u00e3o ser\u00e1 avisado atrav\u00e9s do n\u00famero dentro da raiz: se esse n\u00famero for negativo, ent\u00e3o n\u00e3o ser\u00e1 poss\u00edvel fatorar (com n\u00fameros reais). Tudo isso podemos simplificar introduzindo a conven\u00e7\u00e3o de nota\u00e7\u00e3o:<\/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;\">Que por sua vez se resume na velha e confi\u00e1vel<\/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;\">De modo que a fatora\u00e7\u00e3o finalmente ficar\u00e1 na forma<\/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>Expans\u00e3o para a fatora\u00e7\u00e3o do polin\u00f4mio bi-quadr\u00e1tico<\/h2>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=997s\" target=\"_blank\" rel=\"noopener\"><strong>Esta t\u00e9cnica tamb\u00e9m pode ser usada para fatorar o polin\u00f4mio bi-quadr\u00e1tico<\/strong><\/a> da seguinte forma:<\/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;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\"> x^2_{1,2} = \\displaystyle \\dfrac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. Assim, voc\u00ea pode escrever<\/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;\">Neste ponto, voc\u00ea deve ter cuidado, pois o que vem a seguir tem suas restri\u00e7\u00f5es. Se <span class=\"katex-eq\" data-katex-display=\"false\">x_1^2<\/span> n\u00e3o for um n\u00famero positivo, ent\u00e3o voc\u00ea poder\u00e1 usar soma por diferen\u00e7a para separar <span class=\"katex-eq\" data-katex-display=\"false\">(x^2 - x_1^2) = (x-x_1)(x + x_1)<\/span>; caso contr\u00e1rio, voc\u00ea encontrar\u00e1 n\u00fameros complexos e, portanto, n\u00e3o poder\u00e1 continuar a fatorar em n\u00fameros reais. Se as ra\u00edzes estiverem bem definidas, voc\u00ea poder\u00e1 escrever:<\/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;\">Caso contr\u00e1rio, voc\u00ea parar\u00e1 no passo anterior.<\/p>\n<h3>Generaliza\u00e7\u00e3o para a fatora\u00e7\u00e3o do polin\u00f4mio (2n)-quadr\u00e1tico<\/h3>\n<p style=\"text-align: justify;\"><a href=\"https:\/\/www.youtube.com\/watch?v=ddTfUR7QBfY&amp;t=1521s\" target=\"_blank\" rel=\"noopener\"><strong>Com isso j\u00e1 se entende para onde o m\u00e9todo aponta, para fatorar o polin\u00f4mio (2n)-quadr\u00e1tico<\/strong><\/a> basta reformular a forma como \u00e9 escrito e usar os m\u00e9todos anteriores onde as ra\u00edzes estiverem bem definidas. Desta forma, teremos que:<\/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;\">Onde <span class=\"katex-eq\" data-katex-display=\"false\">x^n_{1,2} =\\displaystyle \\frac{-b \\pm \\sqrt{b^2 - 4ac }}{2a}<\/span>. Em seguida, com isso, separamos com soma por diferen\u00e7a onde n\u00e3o aparecerem n\u00fameros complexos.<\/p>\n<p>   <a name=\"5\"><\/a><\/p>\n<h2>Exerc\u00edcios de exemplo:<\/h2>\n<p style=\"text-align: justify;\">Agora \u00e9 a sua vez de testar essas t\u00e9cnicas com alguns exerc\u00edcios. Os polin\u00f4mios listados a seguir foram escolhidos completamente ao acaso, de modo que ser\u00e3o muito \u00fateis para reconhecer as poss\u00edveis dificuldades que podem surgir ao fatorar essas express\u00f5es.<\/p>\n<h3>Primeiro Round<\/h3>\n<p style=\"text-align: justify;\">Esses polin\u00f4mios s\u00e3o os exemplos que coloquei no in\u00edcio deste 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>Segundo Round<\/h3>\n<p style=\"text-align: justify;\">E aqui est\u00e3o alguns outros um pouco mais dif\u00edceis.<\/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>Solu\u00e7\u00e3o dos Exerc\u00edcios<\/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>Fatora\u00e7\u00e3o do Polin\u00f4mio Quadr\u00e1tico e 2n-Quadr\u00e1tico Resumo: Nesta aula, revisaremos detalhadamente o processo de fatora\u00e7\u00e3o de polin\u00f4mios quadr\u00e1ticos e polin\u00f4mios -quadr\u00e1ticos , decompondo-os em fatores simples. Ser\u00e3o desenvolvidos procedimentos matem\u00e1ticos e exemplos pr\u00e1ticos ser\u00e3o mostrados. Objetivos de aprendizagem Aprender a fatorar polin\u00f4mios quadr\u00e1ticos da forma . Derivar e utilizar a f\u00f3rmula quadr\u00e1tica para encontrar as [&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":3,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-28834","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra-e-geometria","category-matematica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Fatora\u00e7\u00e3o do Polin\u00f4mio Quadr\u00e1tico e 2n-Quadr\u00e1tico - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Descubra como realizar a Fatora\u00e7\u00e3o do Polin\u00f4mio Quadr\u00e1tico de forma simples. 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