{"id":340,"date":"2023-07-06T03:44:20","date_gmt":"2023-07-06T03:44:20","guid":{"rendered":"https:\/\/mathority.org\/pt\/trimestre-binomial\/"},"modified":"2023-07-06T03:44:20","modified_gmt":"2023-07-06T03:44:20","slug":"trimestre-binomial","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/trimestre-binomial\/","title":{"rendered":"Par em quarto"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula de um bin\u00f4mio elevado a quatro, e explicamos como resolver este tipo de opera\u00e7\u00e3o binomial com exemplos. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo pelos colegas at\u00e9 a quarta s\u00e9rie. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formula-del-binomio-a-la-cuarta\"><\/span> F\u00f3rmula binomial trimestral<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Em matem\u00e1tica, um <strong>bin\u00f4mio elevado a quatro<\/strong> \u00e9 um polin\u00f4mio composto por dois termos elevado \u00e0 4\u00aa.<\/p>\n<p> Assim, a f\u00f3rmula utilizada para calcular um trimestre binomial \u00e9 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72f362dcdac7a948e05876758e78e7c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"310\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Esta f\u00f3rmula pode ser derivada da <a href=\"https:\/\/mathority.org\/pt\/formula-do-teorema-binomial-ou-binomial-de-newton-e-exercicios-resolvidos\/\"><strong><span style=\"text-decoration: underline;\">f\u00f3rmula binomial geral de Newton<\/span><\/strong><\/a> . Na verdade, com o bin\u00f4mio de Newton voc\u00ea pode calcular bin\u00f4mios elevados a qualquer pot\u00eancia, ent\u00e3o \u00e9 melhor aprender a f\u00f3rmula binomial de Newton. Clique no link anterior e descubra como \u00e9 essa f\u00f3rmula.<\/p>\n<p> Portanto, um bin\u00f4mio no quarto \u00e9 igual ao primeiro termo elevado ao quarto, mais o produto de 4 vezes o primeiro termo ao cubo e o segundo termo, mais o primeiro e o segundo termos ao quadrado vezes 6, mais o produto de 4 vezes o primeiro termo multiplicado pelo segundo termo elevado a 3, mais o segundo termo elevado a quarto.<\/p>\n<p> Esta f\u00f3rmula corresponde \u00e0 soma binomial (seus dois elementos s\u00e3o positivos), mas na <strong>f\u00f3rmula da subtra\u00e7\u00e3o binomial<\/strong> elevada ao quarto, os sinais do segundo e quarto produtos s\u00e3o negativos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab80a7a5a90fe6612c4474b5d7a8ae6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a \\color{red}\\bm{-}\\color{black}b)^4 = a^4\\color{red}\\bm{-}\\color{black}4a^3b+6a^2b^2\\color{red}\\bm{-}\\color{black}4ab^3+b^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"484\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplos-de-binomios-a-la-cuarta\"><\/span> Exemplos de colegas na quarta s\u00e9rie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Dada a f\u00f3rmula para este tipo de bin\u00f4mio, veremos v\u00e1rios exemplos de resolu\u00e7\u00e3o de um bin\u00f4mio elevado a quatro. Primeiro calcularemos um bin\u00f4mio positivo e depois resolveremos um bin\u00f4mio negativo.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/h3>\n<ul>\n<li> Calcule o seguinte bin\u00f4mio elevado \u00e0 quarta:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-03f5e137fa0eccec573bc15307d33cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula para a pot\u00eancia de uma soma binomial elevada a 4 \u00e9:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72f362dcdac7a948e05876758e78e7c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"310\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, para calcular o bin\u00f4mio do exerc\u00edcio, basta substituir os dois valores do bin\u00f4mio na f\u00f3rmula:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1517da3b83cb1367db7ebf73c8600ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)^4 = x^4+4\\cdot x^3\\cdot 2+6\\cdot x^2\\cdot 2^2+4\\cdot x\\cdot 2^3+2^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"396\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E finalmente resolvemos as opera\u00e7\u00f5es:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4e84ddbeb643abf6cdce05666af6d41c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}(x+2)^4 &amp; = x^4+4\\cdot x^3\\cdot 2+6\\cdot x^2\\cdot 4+4\\cdot x\\cdot 8+16 \\\\[2ex] &amp; =x^4+8 x^3+24x^2+32x+16\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"383\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/h3>\n<ul>\n<li> Encontre o seguinte bin\u00f4mio elevado \u00e0 quarta:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-210f9597e603db7f4e3c41d2d5a1cd9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-3)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula de potencializa\u00e7\u00e3o para um bin\u00f4mio de diferen\u00e7a elevado \u00e0 4\u00aa \u00e9 a seguinte:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d5774034dc5f470b13498faa68571db6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^4 = a^4-4a^3b+6a^2b^2-4ab^3+b^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"310\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, para determinar o bin\u00f4mio do problema, basta substituir as vari\u00e1veis da f\u00f3rmula pelos valores do bin\u00f4mio:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-116893deddf1913174508c08c874b685_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-3)^4 = x^4-4\\cdot x^3\\cdot 3+6\\cdot x^2\\cdot 3^2-4\\cdot x\\cdot 3^3+3^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"396\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E finalmente, resolvemos as opera\u00e7\u00f5es resultantes: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa9fd790c949488c4124c2370fff7ea4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}(x-3)^4 &amp; = x^4-4\\cdot x^3\\cdot 3+6\\cdot x^2\\cdot 9-4\\cdot x\\cdot 27+81 \\\\[2ex] &amp; =x^4-12x^3+54x^2-108x+81\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"391\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demostracion-de-la-formula-de-un-binomio-a-la-cuarta\"><\/span> Demonstra\u00e7\u00e3o da f\u00f3rmula de um bin\u00f4mio na quarta<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para explorar o conceito de bin\u00f4mio elevado \u00e0 quarta, demonstraremos sua f\u00f3rmula de diversas maneiras.<\/p>\n<p> De qualquer par elevado a 4:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-221292a85028b8f8aeec1e8878d7e4d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A express\u00e3o alg\u00e9brica de um bin\u00f4mio elevado \u00e0 quarta pode ser fatorada expandindo-a em fatores primos:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cb4cda88fca732a0609abddf72c7d2ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^4=(a+b)\\cdot (a+b)\\cdot (a+b)\\cdot (a+b)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Assim, resolvendo cada <a href=\"https:\/\/mathority.org\/pt\/multiplicacao-de-polinomios-exemplos-exercicios-produto-resolvido-multiplicacao\/\"><strong><span style=\"text-decoration: underline;\">produto de polin\u00f4mios<\/span><\/strong><\/a> , chegamos \u00e0 f\u00f3rmula do bin\u00f4mio elevado \u00e0 quarta:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1f1fccc8f73eedbccfa9ef436549c932_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a+b)^4 &amp; =(a+b)\\cdot (a+b)\\cdot (a+b)\\cdot (a+b) \\\\[2ex] &amp;= (a^2+2ab+b^2)\\cdot (a+b)\\cdot (a+b) \\\\[2ex] &amp; = (a^3+3a^2b+3ab^2+b^3) \\cdot (a+b) \\\\[2ex] &amp; = a^4+4a^3b+6a^2b^2+4ab^3+b^4 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"150\" width=\"330\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Por outro lado, a f\u00f3rmula de um bin\u00f4mio elevado ao quarto tamb\u00e9m pode ser verificada usando a f\u00f3rmula de um <a href=\"https:\/\/mathority.org\/pt\/binomio-ao-cubo\/\"><strong><span style=\"text-decoration: underline;\">bin\u00f4mio elevado ao cubo<\/span><\/strong><\/a> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b87645d508e38be4981e6cd954490f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a+b)^4 &amp; =(a+b)^3 \\cdot (a+b)\\\\[2ex] &amp; = (a^3+3a^2b+3ab^2+b^3) \\cdot (a+b) \\\\[2ex] &amp; = a^4+4a^3b+6a^2b^2+4ab^3+b^4 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"326\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Da mesma forma, a prova pode ser obtida atrav\u00e9s de produtos not\u00e1veis (ou identidades not\u00e1veis). Por exemplo, usando a f\u00f3rmula do <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/quadrado-de-uma-soma-ou-soma-ao-quadrado\/\">produto not\u00e1vel do quadrado de uma soma<\/a><\/span><\/strong> :<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f6a8ef4bfb728886326da5d4fd39974a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a+b)^4 &amp; =(a+b)^2\\cdot (a+b)^2 \\\\[2ex] &amp;= (a^2+2ab+b^2)\\cdot (a^2+2ab+b^2) \\\\[2ex]  &amp; = a^4+4a^3b+6a^2b^2+4ab^3+b^4 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"325\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Respectivamente, a not\u00e1vel f\u00f3rmula de identidade para o <a href=\"https:\/\/mathority.org\/pt\/quadrado-de-uma-diferenca-ou-subtracao\/\"><strong><span style=\"text-decoration: underline;\">quadrado de uma subtra\u00e7\u00e3o<\/span><\/strong><\/a> \u00e9 usada para corroborar a f\u00f3rmula para uma subtra\u00e7\u00e3o binomial: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-14234e8714d33a06a9c2b029fec252d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a-b)^4 &amp; =(a-b)^2\\cdot (a-b)^2 \\\\[2ex] &amp;= (a^2-2ab+b^2)\\cdot (a^2-2ab+b^2) \\\\[2ex]  &amp; = a^4-4a^3b+6a^2b^2-4ab^3+b^4 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"325\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-binomios-a-la-cuarta\"><\/span> Exerc\u00edcios resolvidos para colegas da quarta s\u00e9rie<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Resolva as seguintes pot\u00eancias de bin\u00f4mios elevados \u00e0 quarta: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-78de0250a24376eafa33b75828530e55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ (x+1)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c55ffad970f894d6ceb6951057bfe5aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ (2x+3)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fdd88fec0f06da304ff2999b733fc313_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ (x-4)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"87\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-33170b079c2a7a10373f56099500b550_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ (x^2+y)^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"96\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__DDF5FF\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#DDF5FF\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>veja solu\u00e7\u00e3o<\/strong> <\/div>\n<\/div>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-063eb752a20c15adfc0eec5a42b9d67e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned} (x+1)^4 &amp; = x^4 +4\\cdot x^3\\cdot 1+6 \\cdot x^2\\cdot 1^2+4 \\cdot x\\cdot 1^3 + 1^4 \\\\[2ex]  &amp; = x^4 +4\\cdot x^3\\cdot 1+6 \\cdot x^2\\cdot 1+4 \\cdot x\\cdot 1 + 1 \\\\[2ex]  &amp; = \\bm{x^4 +4x^3+6 x^2+4 x + 1}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"429\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-707c85ccb35055d4fece1d058b46217a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned} (2x+3)^4 &amp; = (2x)^4 +4\\cdot (2x)^3\\cdot 3+6 \\cdot (2x)^2\\cdot 3^2+4 \\cdot 2x\\cdot 3^3 + 3^4 \\\\[2ex]  &amp; = 16x^4 +4\\cdot 8x^3\\cdot 3+6 \\cdot 4x^2\\cdot 9+4 \\cdot 2x\\cdot 27 + 81\\\\[2ex]  &amp; = \\bm{16x^4 +96x^3+216x^2+216x + 81}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"514\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce610d6757b7ce6fb5dfaa3b0679f368_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned} (x-4)^4 &amp; = x^4 -4\\cdot x^3\\cdot 4+6 \\cdot x^2\\cdot 4^2-4 \\cdot x\\cdot 4^3 + 4^4 \\\\[2ex]  &amp; = x^4 -4\\cdot x^3\\cdot 4+6 \\cdot x^2\\cdot 16-4 \\cdot x\\cdot 64 + 256 \\\\[2ex] &amp; = \\bm{x^4 -16 x^3+96x^2-256x + 256}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"442\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4f5ca1d6b47e6ed78f64deff8122ed3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned} (x^2+y)^4 &amp; = \\left(x^2\\right)^4 +4\\cdot \\left(x^2\\right)^3\\cdot y+6 \\cdot \\left(x^2\\right)^2\\cdot y^2+4 \\cdot x^2\\cdot y^3 + y^4 \\\\[2ex] &amp; =x^8 +4\\cdot x^6\\cdot y+6 \\cdot x^4\\cdot y^2+4 \\cdot x^2\\cdot y^3 + y^4 \\\\[2ex]  &amp; = \\bm{x^8 +4x^6y+6 x^4 y^2+4x^2y^3 + y^4}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"112\" width=\"519\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula de um bin\u00f4mio elevado a quatro, e explicamos como resolver este tipo de opera\u00e7\u00e3o binomial com exemplos. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo pelos colegas at\u00e9 a quarta s\u00e9rie. F\u00f3rmula binomial trimestral Em matem\u00e1tica, um bin\u00f4mio elevado a quatro \u00e9 um polin\u00f4mio composto por &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\"> <span class=\"screen-reader-text\">Par em quarto<\/span> Leia mais &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[7],"tags":[],"class_list":["post-340","post","type-post","status-publish","format-standard","hentry","category-binomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Bin\u00f4mio quarto - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Bin\u00f4mio quarto - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula de um bin\u00f4mio elevado a quatro, e explicamos como resolver este tipo de opera\u00e7\u00e3o binomial com exemplos. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo pelos colegas at\u00e9 a quarta s\u00e9rie. F\u00f3rmula binomial trimestral Em matem\u00e1tica, um bin\u00f4mio elevado a quatro \u00e9 um polin\u00f4mio composto por &hellip; Par em quarto Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T03:44:20+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72f362dcdac7a948e05876758e78e7c1_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Par em quarto\",\"datePublished\":\"2023-07-06T03:44:20+00:00\",\"dateModified\":\"2023-07-06T03:44:20+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\"},\"wordCount\":543,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Bin\u00f4mios\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/trimestre-binomial\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\",\"url\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\",\"name\":\"Bin\u00f4mio quarto - Matoridade\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/#website\"},\"datePublished\":\"2023-07-06T03:44:20+00:00\",\"dateModified\":\"2023-07-06T03:44:20+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/#breadcrumb\"},\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/mathority.org\/pt\/trimestre-binomial\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/mathority.org\/pt\/trimestre-binomial\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/mathority.org\/pt\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Par em quarto\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/mathority.org\/pt\/#website\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"name\":\"Mathority\",\"description\":\"Onde a curiosidade encontra o c\u00e1lculo!\",\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/mathority.org\/pt\/?s={search_term_string}\"},\"query-input\":\"required name=search_term_string\"}],\"inLanguage\":\"pt-BR\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/mathority.org\/pt\/#organization\",\"name\":\"Mathority\",\"url\":\"https:\/\/mathority.org\/pt\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"contentUrl\":\"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png\",\"width\":703,\"height\":151,\"caption\":\"Mathority\"},\"image\":{\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\",\"name\":\"Equipe Mathoridade\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"pt-BR\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g\",\"caption\":\"Equipe Mathoridade\"},\"sameAs\":[\"http:\/\/mathority.org\/pt\"]}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Bin\u00f4mio quarto - Matoridade","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/trimestre-binomial\/","og_locale":"pt_BR","og_type":"article","og_title":"Bin\u00f4mio quarto - Matoridade","og_description":"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula de um bin\u00f4mio elevado a quatro, e explicamos como resolver este tipo de opera\u00e7\u00e3o binomial com exemplos. Al\u00e9m disso, voc\u00ea poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo pelos colegas at\u00e9 a quarta s\u00e9rie. F\u00f3rmula binomial trimestral Em matem\u00e1tica, um bin\u00f4mio elevado a quatro \u00e9 um polin\u00f4mio composto por &hellip; Par em quarto Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/trimestre-binomial\/","article_published_time":"2023-07-06T03:44:20+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-72f362dcdac7a948e05876758e78e7c1_l3.png"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"3 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Par em quarto","datePublished":"2023-07-06T03:44:20+00:00","dateModified":"2023-07-06T03:44:20+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/"},"wordCount":543,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Bin\u00f4mios"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/trimestre-binomial\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/","url":"https:\/\/mathority.org\/pt\/trimestre-binomial\/","name":"Bin\u00f4mio quarto - Matoridade","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/#website"},"datePublished":"2023-07-06T03:44:20+00:00","dateModified":"2023-07-06T03:44:20+00:00","breadcrumb":{"@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/#breadcrumb"},"inLanguage":"pt-BR","potentialAction":[{"@type":"ReadAction","target":["https:\/\/mathority.org\/pt\/trimestre-binomial\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/mathority.org\/pt\/trimestre-binomial\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/mathority.org\/pt\/"},{"@type":"ListItem","position":2,"name":"Par em quarto"}]},{"@type":"WebSite","@id":"https:\/\/mathority.org\/pt\/#website","url":"https:\/\/mathority.org\/pt\/","name":"Mathority","description":"Onde a curiosidade encontra o c\u00e1lculo!","publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/mathority.org\/pt\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"pt-BR"},{"@type":"Organization","@id":"https:\/\/mathority.org\/pt\/#organization","name":"Mathority","url":"https:\/\/mathority.org\/pt\/","logo":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/","url":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","contentUrl":"https:\/\/mathority.org\/pt\/wp-content\/uploads\/2023\/10\/mathority-logo.png","width":703,"height":151,"caption":"Mathority"},"image":{"@id":"https:\/\/mathority.org\/pt\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00","name":"Equipe Mathoridade","image":{"@type":"ImageObject","inLanguage":"pt-BR","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/8a35e4c8616d1c34c03ca02862b580f4372c5650665668489db53a09579bbc4f?s=96&d=mm&r=g","caption":"Equipe Mathoridade"},"sameAs":["http:\/\/mathority.org\/pt"]}]}},"yoast_meta":{"yoast_wpseo_title":"","yoast_wpseo_metadesc":"","yoast_wpseo_canonical":""},"_links":{"self":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/340","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/comments?post=340"}],"version-history":[{"count":0,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/posts\/340\/revisions"}],"wp:attachment":[{"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/media?parent=340"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/categories?post=340"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathority.org\/pt\/wp-json\/wp\/v2\/tags?post=340"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}