{"id":27137,"date":"2021-03-29T13:00:55","date_gmt":"2021-03-29T13:00:55","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27137"},"modified":"2024-06-15T08:47:01","modified_gmt":"2024-06-15T08:47:01","slug":"algebra-de-polinomios-de-numeros-reais","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/","title":{"rendered":"\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px; text-align:center;\">\n<h1>\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nNesta aula, exploraremos a \u00e1lgebra de polin\u00f4mios, sua defini\u00e7\u00e3o, propriedades e aplica\u00e7\u00f5es. Os polin\u00f4mios s\u00e3o uma parte fundamental da matem\u00e1tica e t\u00eam amplas aplica\u00e7\u00f5es em diversas disciplinas.<br \/>\n    <\/em><\/p>\n<p>    <strong>OBJETIVOS DE APRENDIZAGEM<\/strong><\/p>\n<p>Ao finalizar esta aula, o estudante ser\u00e1 capaz de:<\/p>\n<p style=\"text-align:left;\">\n        1. Definir e compreender os polin\u00f4mios e suas propriedades.<br \/>\n        2. Identificar o grau e os coeficientes de um polin\u00f4mio.<br \/>\n        3. Realizar opera\u00e7\u00f5es alg\u00e9bricas com polin\u00f4mios e aplicar suas propriedades em contextos matem\u00e1ticos.\n    <\/p>\n<p>    <strong>\u00cdNDICE DE CONTE\u00daDOS:<\/strong><\/p>\n<p>\n        <a href=\"#1\"><strong>1. \u00c1lgebra de Polin\u00f4mios: Defini\u00e7\u00f5es<\/strong><\/a><br \/>\n        <a href=\"#2\"><strong>2. Tipos de Polin\u00f4mios<\/strong><\/a><br \/>\n        <a href=\"#3\"><strong>3. \u00c1lgebra de Polin\u00f4mios: Opera\u00e7\u00f5es<\/strong><\/a><br \/>\n        <a href=\"#4\"><strong>4. Fatora\u00e7\u00e3o e Divis\u00e3o de Polin\u00f4mios<\/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. \u00c1lgebra de Polin\u00f4mios: Defini\u00e7\u00f5es<\/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;\">Para entender a \u00c1lgebra de Polin\u00f4mios, primeiro precisamos saber o que s\u00e3o polin\u00f4mios.<\/span><\/strong><\/a> Os polin\u00f4mios s\u00e3o fun\u00e7\u00f5es alg\u00e9bricas. Se <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> \u00e9 uma vari\u00e1vel real, ent\u00e3o dizemos que a fun\u00e7\u00e3o <span class=\"katex-eq\" data-katex-display=\"false\">P(x)<\/span> \u00e9 um polin\u00f4mio se puder ser escrita na forma:\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    onde <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> \u00e9 um inteiro n\u00e3o negativo e todos os <span class=\"katex-eq\" data-katex-display=\"false\">a_i<\/span>, com <span class=\"katex-eq\" data-katex-display=\"false\">i\\in\\{1,2,3,\\cdots,n\\},<\/span> s\u00e3o coeficientes reais. Se existe um <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> tal que <span class=\"katex-eq\" data-katex-display=\"false\">a_k\\neq 0<\/span> e, quando <span class=\"katex-eq\" data-katex-display=\"false\">k\\lt i<\/span>, ocorre que <span class=\"katex-eq\" data-katex-display=\"false\">a_i=0<\/span>, ent\u00e3o dizemos que tal valor de <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u00e9 <strong>o grau do polin\u00f4mio.<\/strong> Em outras palavras, o grau de um polin\u00f4mio \u00e9 a pot\u00eancia maior que acompanha um coeficiente diferente de zero.\n<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2><strong>2. Tipos de Polin\u00f4mios<\/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;\">Os polin\u00f4mios s\u00e3o classificados de acordo com seu grau;<\/span><\/strong><\/a> por isso, quando se menciona um polin\u00f4mio, quase sempre se diz que \u00e9 um polin\u00f4mio de grau <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span>, quando <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> \u00e9 a maior pot\u00eancia de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> que acompanha o coeficiente n\u00e3o nulo de tal polin\u00f4mio.\n<\/p>\n<h3>2.1. Os Polin\u00f4mios Constantes<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u00c9 a fam\u00edlia que engloba todos os polin\u00f4mios de grau zero e o polin\u00f4mio nulo. Dizemos que um polin\u00f4mio \u00e9 de grau zero, se puder ser escrito na forma <span class=\"katex-eq\" data-katex-display=\"false\">P(x)=c,<\/span> com <span class=\"katex-eq\" data-katex-display=\"false\">c\\neq 0.<\/span> Por outro lado, o polin\u00f4mio nulo \u00e9 da forma <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 0<\/span> e para este n\u00e3o se define um grau.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2><strong>3. \u00c1lgebra de Polin\u00f4mios: Opera\u00e7\u00f5es<\/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;\">Os polin\u00f4mios herdam todas as suas propriedades da \u00e1lgebra dos n\u00fameros reais.<\/span><\/strong><\/a> S\u00e3o especialmente relevantes as propriedades distributivas e associativas.\n<\/p>\n<h3>3.1. Soma e Subtra\u00e7\u00e3o<\/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;\">Se <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">Q<\/span> s\u00e3o dois polin\u00f4mios de grau<\/span><\/strong><\/a> <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">m<\/span>, respectivamente, com<\/p>\n<p style=\"text-align:center\"><span class=\"katex-eq\" data-katex-display=\"false\">m=n+k<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">0\\leq k,<\/span>\n<p style=\"text-align: justify; color: #000000;\">ent\u00e3o ter\u00e1 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;\">Ou seja, os coeficientes que acompanham as mesmas pot\u00eancias de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> s\u00e3o somados ou subtra\u00eddos, conforme o caso.<\/p>\n<p style=\"text-align: justify; color: #000000;\"><span style=\"color: #000080;\">EXEMPLO:<\/span><br \/>\nSe <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 3+5x+2x^2<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 6x-3x^2 +23x^5<\/span>, ent\u00e3o:<\/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. Multiplica\u00e7\u00e3o<\/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;\">No mesmo contexto da soma e subtra\u00e7\u00e3o de polin\u00f4mios,<\/span><\/strong><\/a> o produto de polin\u00f4mios ser\u00e1 desenvolvido da seguinte forma:\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Primeiro distinguimos a multiplica\u00e7\u00e3o por escalar. Se <span class=\"katex-eq\" data-katex-display=\"false\">c \\in \\mathbb{R},<\/span> ent\u00e3o temos:\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    E depois temos a multiplica\u00e7\u00e3o entre polin\u00f4mios:\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    Isto \u00e9 o que resumir\u00edamos atrav\u00e9s da express\u00e3o \u00aba soma dos produtos de todos com todos\u00bb.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXEMPLO:<\/span><br \/>\n    Se <span class=\"katex-eq\" data-katex-display=\"false\">P(x) = 4x+ 2x^2-x^4<\/span> e <span class=\"katex-eq\" data-katex-display=\"false\">Q(x) = 5 - x + x^2-7x^3,<\/span> ent\u00e3o:\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. Fatora\u00e7\u00e3o e Divis\u00e3o de Polin\u00f4mios<\/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;\">Quando multiplicamos dois polin\u00f4mios, o que fazemos \u00e9 passar de dois polin\u00f4mios simples para outro mais complicado (de maior grau).<\/span><\/strong><\/a> Quando fatoramos um polin\u00f4mio seguimos o processo inverso: transformamos um polin\u00f4mio complicado no produto de dois ou mais polin\u00f4mios de menor grau.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    Para fatorar um polin\u00f4mio <span class=\"katex-eq\" data-katex-display=\"false\">P(x),<\/span> \u00e9 necess\u00e1rio encontrar os valores de <span class=\"katex-eq\" data-katex-display=\"false\">x<\/span> que anulam o polin\u00f4mio; se tais valores existirem, ent\u00e3o o polin\u00f4mio \u00e9 fator\u00e1vel. Falar de exist\u00eancia \u00e9 acess\u00edvel, mas falar de encontr\u00e1-los \u00e9 uma hist\u00f3ria diferente. Revisaremos este tema com mais detalhe quando estudarmos as fatora\u00e7\u00e3os dos polin\u00f4mios quadr\u00e1ticos e (2n)quadr\u00e1ticos.\n<\/p>\n<h3>4.1. Produtos Not\u00e1veis<\/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;\">Existem, no entanto, casos em que a fatora\u00e7\u00e3o \u00e9 obtida de um modo simples,<\/span><\/strong><br \/>\n    <\/a> como o dos produtos not\u00e1veis. Alguns desses resultados s\u00e3o os seguintes:\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>4.2. O Algoritmo da divis\u00e3o<\/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;\">Assim como multiplicando inteiros obtemos n\u00fameros compostos e a divis\u00e3o atrav\u00e9s do algoritmo da divis\u00e3o nos permite fatorar quando o resto \u00e9 zero,<\/span><\/strong><\/a> algo semelhante ocorre com os polin\u00f4mios. Explicar o algoritmo da divis\u00e3o \u00abem texto\u00bb pode ser um pouco complicado, \u00e9 muito mais f\u00e1cil de entender vendo diretamente como se faz e em que casos o algoritmo conduz a uma fatora\u00e7\u00e3o. Para isso, revisaremos alguns exemplos.\n<\/p>\n<p style=\"text-align: justify; color: #000000;\">\n    <span style=\"color: #000080;\">EXEMPLO:<\/span> Calcular <span class=\"katex-eq\" data-katex-display=\"false\">P(x):Q(x)<\/span> para os seguintes casos:\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;\">[SOLU\u00c7\u00c3O]<\/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>[SOLU\u00c7\u00c3O]<\/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>[SOLU\u00c7\u00c3O]<\/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>[SOLU\u00c7\u00c3O]<\/strong><\/span> <\/a>\n    <\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais Resumo: Nesta aula, exploraremos a \u00e1lgebra de polin\u00f4mios, sua defini\u00e7\u00e3o, propriedades e aplica\u00e7\u00f5es. Os polin\u00f4mios s\u00e3o uma parte fundamental da matem\u00e1tica e t\u00eam amplas aplica\u00e7\u00f5es em diversas disciplinas. OBJETIVOS DE APRENDIZAGEM Ao finalizar esta aula, o estudante ser\u00e1 capaz de: 1. Definir e compreender os polin\u00f4mios e suas propriedades. [&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":3,"footnotes":""},"categories":[587,571],"tags":[],"class_list":["post-27137","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>\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda sobre \u00e1lgebra de polin\u00f4mios: defini\u00e7\u00f5es, tipos, opera\u00e7\u00f5es, produtos, fatora\u00e7\u00e3o e divis\u00e3o.\" \/>\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\/pt\/algebra-de-polinomios-de-numeros-reais\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais\" \/>\n<meta property=\"og:description\" content=\"Aprenda sobre \u00e1lgebra de polin\u00f4mios: defini\u00e7\u00f5es, tipos, opera\u00e7\u00f5es, produtos, fatora\u00e7\u00e3o e divis\u00e3o.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/\" \/>\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=\"2021-03-29T13:00:55+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-15T08:47:01+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=\"\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais\" \/>\n<meta name=\"twitter:description\" content=\"Aprenda sobre \u00e1lgebra de polin\u00f4mios: defini\u00e7\u00f5es, tipos, opera\u00e7\u00f5es, produtos, fatora\u00e7\u00e3o e divis\u00e3o.\" \/>\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\/pt\/algebra-de-polinomios-de-numeros-reais\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais\",\"datePublished\":\"2021-03-29T13:00:55+00:00\",\"dateModified\":\"2024-06-15T08:47:01+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/\"},\"wordCount\":1260,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/03\/polinomios.jpg\",\"articleSection\":[\"\u00c1lgebra e Geometria\",\"Matem\u00e1tica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/\",\"url\":\"http:\/\/toposuranos.com\/material\/pt\/algebra-de-polinomios-de-numeros-reais\/\",\"name\":\"\u00c1lgebra de Polin\u00f4mios de N\u00fameros Reais - 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