{"id":351,"date":"2023-07-06T00:53:57","date_gmt":"2023-07-06T00:53:57","guid":{"rendered":"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/"},"modified":"2023-07-06T00:53:57","modified_gmt":"2023-07-06T00:53:57","slug":"identidades-produtos-igualdades-notaveis-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/","title":{"rendered":"Identidades not\u00e1veis (ou produtos not\u00e1veis)"},"content":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o da resolu\u00e7\u00e3o de todos os tipos de identidades not\u00e1veis (ou produtos not\u00e1veis). Voc\u00ea poder\u00e1 ver quais s\u00e3o as f\u00f3rmulas de todas as identidades not\u00e1veis, bem como exemplos e exerc\u00edcios resolvidos passo a passo. Al\u00e9m disso, mostraremos para que servem essas famosas regras matem\u00e1ticas.<\/p>\n<p> \ud83d\udc49\ud83d\udc49 Abaixo explicamos cada identidade not\u00e1vel passo a passo, mas se preferir pode ir direto para a <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\">tabela \ud83d\ude09 onde todas as f\u00f3rmulas est\u00e3o resumidas<\/a><\/span><\/strong> . \ud83d\udc48\ud83d\udc48 <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-son-las-identidades-notables-o-productos-notables\"><\/span> O que s\u00e3o identidades not\u00e1veis (ou produtos not\u00e1veis)?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Identidades not\u00e1veis<\/strong> , tamb\u00e9m chamadas <strong>de produtos not\u00e1veis<\/strong> ou <strong>igualdades not\u00e1veis<\/strong> , s\u00e3o regras matem\u00e1ticas que permitem resolver diretamente opera\u00e7\u00f5es com polin\u00f4mios.<\/p>\n<p> As f\u00f3rmulas de identidade not\u00e1veis mais comuns s\u00e3o o <em>quadrado de uma soma<\/em> , o <em>quadrado de uma diferen\u00e7a (ou subtra\u00e7\u00e3o)<\/em> e a <em>soma vezes a diferen\u00e7a<\/em> . <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/identites-produits-ou-egalites-notables.png\" alt=\"identidades ou igualdades de produtos not\u00e1veis\" class=\"wp-image-2751\" width=\"281\" height=\"281\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Mas a seguir n\u00e3o apenas ensinaremos como calcular esses produtos not\u00e1veis, mas tamb\u00e9m mostraremos todos os tipos de identidades not\u00e1veis que existem. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formulas-de-las-identidades-o-productos-notables\"><\/span> F\u00f3rmulas (ou produtos) de identidade not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos a defini\u00e7\u00e3o de produtos not\u00e1veis (ou igualdades not\u00e1veis), veremos quais s\u00e3o as f\u00f3rmulas para identidades not\u00e1veis. Por outro lado, se voc\u00ea estiver interessado em demonstra\u00e7\u00f5es de f\u00f3rmulas, poder\u00e1 visualiz\u00e1-las clicando nos bot\u00f5es \u201cver demonstra\u00e7\u00e3o\u201d.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cuadrado-de-una-suma\"><\/span> quadrado de uma soma<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O <strong>quadrado de uma soma<\/strong> , ou <strong>soma ao quadrado<\/strong> , \u00e9 uma das principais identidades not\u00e1veis. Mais precisamente, \u00e9 um bin\u00f4mio com dois termos positivos elevado a 2, ou seja, sua express\u00e3o alg\u00e9brica \u00e9 <strong>(a+b) <sup>2<\/sup><\/strong> .<\/p>\n<p> Portanto, a f\u00f3rmula do quadrado de uma soma \u00e9: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-pour-le-carre-dune-somme.png\" alt=\"identidades not\u00e1veis ao quadrado com uma soma\" class=\"wp-image-2339\" width=\"274\" height=\"275\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Se partirmos de um bin\u00f4mio positivo elevado a 2:<\/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-ef98ef741811c17cd99e75e5f848ea69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Matematicamente, o quadrado acima \u00e9 equivalente ao fator<\/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-2cfde4aacc08aae1ed70fe5f7b2f74de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> multiplicado por si mesmo:<\/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-7b2df207ac593eaf04ac60ac40b89a7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2=(a+b)\\cdot (a+b)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"200\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, multiplicamos polin\u00f4mios usando a propriedade distributiva:<\/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-1c871c4ad6546c817128379acbef78c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a+b)\\cdot (a+b) &amp; = a\\cdot a +a\\cdot b +b\\cdot a +b\\cdot b \\\\[2ex] &amp;=a^2+ab+ba+b^2 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"325\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Dos quatro termos obtidos,<\/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-7202c73e2795274765d7f01eefc3e3f5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"ab\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e095edd42169777a1290a880eecae4ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"ba\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> parecem semelhantes para que possamos agrup\u00e1-los:<\/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-b645fb320040c599e077b3e5bdc4b407_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2+ab+ba+b^2 = a^2+2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"256\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Tanto que j\u00e1 chegamos \u00e0 express\u00e3o da f\u00f3rmula da soma quadrada, da qual ela \u00e9 derivada:<\/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-66c4071b50f376018a8ac9b6f3f9f5fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2= a^2+2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A t\u00edtulo de curiosidade, o desenvolvimento da express\u00e3o para este tipo de produto not\u00e1vel \u00e9 denominado trin\u00f4mio quadrado perfeito.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> De modo que o quadrado de uma soma \u00e9 igual ao quadrado do primeiro termo, mais o dobro do produto do primeiro pelo segundo, mais o quadrado do segundo.<\/p>\n<p> Portanto, para resolver uma soma quadrada, n\u00e3o basta elevar cada adi\u00e7\u00e3o a ambas, mas, al\u00e9m disso, as duas adi\u00e7\u00f5es devem ser multiplicadas entre si e por 2. \u00c9 importante lembrar disso porque um erro muito t\u00edpico deste tipo do produto \u00c9 not\u00e1vel esquecer este termo. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/polynomes-au-carre-dune-somme.jpg\" alt=\"identidades polinomiais e binomiais not\u00e1veis\" class=\"wp-image-2342\" width=\"246\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h4 class=\"wp-block-heading\"> Exemplo:<\/h4>\n<ul>\n<li> Calcule a seguinte identidade not\u00e1vel aplicando sua f\u00f3rmula correspondente:<\/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-9a9b2aa575ef7fa83e8bb98cbb385ac8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+5)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como acabamos de ver, a f\u00f3rmula para a igualdade not\u00e1vel de uma soma quadrada \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-e3c7bb69fbb939444db4e075615462f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2=a^2+2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, devemos primeiro identificar os par\u00e2metros<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula. Nesse caso,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> representa o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> do par e<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponde ao n\u00famero 5:<\/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-5ba75b0f34f956985ea0163011a03acf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} (a+b)^2\\\\[2ex] (x+5)^2 \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=x \\\\[2ex] b=5 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o, agora que sabemos os valores de<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a521efcd0a946cd643aebe98b5b41a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> podemos usar a f\u00f3rmula de um bin\u00f4mio quadrado positivo para encontrar o resultado: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/produits-notables-au-carre-dune-somme.jpg\" alt=\"exemplos de identidades quadradas not\u00e1veis\" class=\"wp-image-2349\" width=\"284\" height=\"164\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cuadrado-de-una-diferencia\"><\/span> quadrado de uma diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O <strong>quadrado de uma diferen\u00e7a<\/strong> , ou <strong>diferen\u00e7a ao quadrado<\/strong> , \u00e9 outra das 3 identidades not\u00e1veis mais utilizadas. Em particular, corresponde a um bin\u00f4mio formado por um termo positivo e outro negativo elevado a 2, ou seja, sua express\u00e3o alg\u00e9brica \u00e9 <strong>(ab) <sup>2<\/sup><\/strong> .<\/p>\n<p> Portanto, a f\u00f3rmula do quadrado de uma diferen\u00e7a (ou quadrado de uma subtra\u00e7\u00e3o) \u00e9 a seguinte: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-du-carre-dune-difference-ou-soustraction.png\" alt=\"produtos not\u00e1veis ao quadrado\" class=\"wp-image-2407\" width=\"306\" height=\"307\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Da express\u00e3o binomial de uma subtra\u00e7\u00e3o quadrada:<\/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-8f88949b2f3fcc20e9d00f495e471cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Obviamente, a pot\u00eancia anterior \u00e9 igual ao produto do fator<\/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-5ab5e2adaf0a63382c066ea55b51147c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> multiplicado por si mesmo:<\/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-5786e5179724339feaef50ccdb33ead1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^2= (a-b)\\cdot (a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"200\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora multiplicamos os dois par\u00eanteses aplicando a propriedade distributiva:<\/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-3b46073fd758d93fff8956f0a8dd57af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}(a-b)\\cdot (a-b) &amp; = a\\cdot a +a\\cdot (-b) - b\\cdot a - b \\cdot (-b) \\\\[2ex] &amp; = a^2-ab-ba+b^2 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"379\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o voc\u00ea s\u00f3 precisa agrupar termos semelhantes para finalizar a verifica\u00e7\u00e3o da 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-68a1e26dd180891fc1ee31584a471ca9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-ab-ba+b^2 = a^2-2ab +b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"256\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o a f\u00f3rmula do quadrado de uma diferen\u00e7a \u00e9 provada matematicamente: <\/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-3074d6e8bc69734f38234657d1fddc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^2=a^2-2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> De modo que o quadrado de uma diferen\u00e7a \u00e9 igual ao quadrado do primeiro termo, menos o dobro do produto do primeiro pelo segundo, mais o quadrado do segundo.<\/p>\n<p> Quanto \u00e0 not\u00e1vel igualdade da soma ao quadrado, n\u00e3o devemos esquecer de colocar o termo m\u00e9dio da f\u00f3rmula, pois a seguinte equa\u00e7\u00e3o est\u00e1 incorreta: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/carre-du-binome-d-une-soustraction.jpg\" alt=\"erros comuns de identidade not\u00e1veis\" class=\"wp-image-2409\" width=\"246\" height=\"68\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h4 class=\"wp-block-heading\"> Exemplo:<\/h4>\n<ul>\n<li> Resolva a seguinte igualdade not\u00e1vel de uma diferen\u00e7a quadrada:<\/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-c639502958a3e7b758e74eda141cd322_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-3)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> \u00c9 o produto not\u00e1vel de uma subtra\u00e7\u00e3o ao quadrado, portanto \u00e9 necess\u00e1rio aplicar sua f\u00f3rmula correspondente:<\/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-3074d6e8bc69734f38234657d1fddc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^2=a^2-2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A seguir, devemos identificar quais s\u00e3o os valores das inc\u00f3gnitas.<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula. Nesse caso,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 a vari\u00e1vel<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponde ao n\u00famero 3:<\/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-1bb2d14a30d2cdabae6458f5df32392a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} (a-b)^2\\\\[2ex] (x-3)^2 \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=x \\\\[2ex] b=3 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Observe que o sinal negativo n\u00e3o faz parte do par\u00e2metro<\/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-a521efcd0a946cd643aebe98b5b41a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> mas voc\u00ea deve sempre pegar o n\u00famero sem o sinal para aplicar corretamente esta f\u00f3rmula.<\/p>\n<p> Portanto, j\u00e1 conhecemos os valores de<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> , \u00e9 portanto suficiente substituir esses valores na f\u00f3rmula para resolver a identidade not\u00e1vel: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/binome-dune-soustraction-au-carre-exercices-resolus.jpg\" alt=\"exemplos e exerc\u00edcios resolvidos passo a passo de igualdades not\u00e1veis\" class=\"wp-image-2417\" width=\"299\" height=\"173\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Suma-por-diferencia\"><\/span> soma por diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O <strong>produto de uma soma e uma diferen\u00e7a<\/strong> \u00e9 uma das 3 identidades not\u00e1veis mais utilizadas. Como o pr\u00f3prio nome sugere, \u00e9 um bin\u00f4mio positivo multiplicado pelo seu bin\u00f4mio conjugado (mesmo bin\u00f4mio mas com o sinal intermedi\u00e1rio alterado), ou seja, a express\u00e3o alg\u00e9brica deste tipo de produto not\u00e1vel \u00e9 <strong>(a +b) \u00b7 (ab)<\/strong> .<\/p>\n<p> A f\u00f3rmula para a identidade not\u00e1vel do produto de uma soma por uma diferen\u00e7a \u00e9 a seguinte: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/produit-de-la-somme-par-la-difference.png\" alt=\"identidades, produtos e igualdades not\u00e1veis das escolas secund\u00e1rias 2, 3 e 4 que\" class=\"wp-image-2278\" width=\"247\" height=\"248\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Come\u00e7ando pelo produto de uma soma pela subtra\u00e7\u00e3o de dois termos quaisquer:<\/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-226e878dd6855cddf50e1bd6eeed0eab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\\cdot (a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Para demonstrar a f\u00f3rmula, basta multiplicar o primeiro par\u00eantese pelo segundo par\u00eantese usando a propriedade distributiva:<\/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-346d3d7ca4da1e71fad52c84a33ef4fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{l}(a+b)\\cdot (a-b)= \\\\[2ex] = a\\cdot a +a\\cdot (-b) +b \\cdot a +b\\cdot (-b) =\\\\[2ex] = a^2 -ab+ba-b^2\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"91\" width=\"276\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora agrupamos termos semelhantes:<\/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-cc83a078573a59dfd63c1a7cdad77e01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2 -ab+ba-b^2=a^2-b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"209\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E conseguimos assim a express\u00e3o de uma igualdade not\u00e1vel. Assim \u00e9 demonstrada a f\u00f3rmula para este not\u00e1vel tipo de identidade: <\/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-e1868f84409086d4b0b21464e4a4f207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\\cdot (a-b) =a^2-b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Assim, o produto da soma pela diferen\u00e7a de duas grandezas \u00e9 igual \u00e0 diferen\u00e7a dos quadrados dessas grandezas. Ou, por outras palavras, multiplicar a soma de dois termos diferentes subtraindo esses mesmos dois termos \u00e9 equivalente a elevar ao quadrado cada um dos 2 termos e subtra\u00ed-los.<\/p>\n<h4 class=\"wp-block-heading\"> Exemplo:<\/h4>\n<ul>\n<li> Encontre, usando a f\u00f3rmula correspondente, o seguinte produto not\u00e1vel da soma pela diferen\u00e7a de dois termos diferentes:<\/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-4306df75f7e6d774f71a001d93d0a830_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)\\cdot (x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como vimos acima, a f\u00f3rmula para a igualdade not\u00e1vel de uma soma multiplicada por uma diferen\u00e7a \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-e1868f84409086d4b0b21464e4a4f207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\\cdot (a-b) =a^2-b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiramente o que precisamos fazer \u00e9 identificar os valores das letras<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula. Nesse caso<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponde \u00e0 vari\u00e1vel<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponde ao n\u00famero 2.<\/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-87b76b09924467ba75f033336e6a18e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} (a+b)\\cdot (a-b) \\\\[2ex] (x+2)\\cdot (x-2) \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=x \\\\[2ex] b=2 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"355\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E quando j\u00e1 sabemos quais valores os par\u00e2metros assumem<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a521efcd0a946cd643aebe98b5b41a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> Aplicamos a f\u00f3rmula do produto da soma pela diferen\u00e7a: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cuadrado-de-un-trinomio\"><\/span>quadrado de um trin\u00f4mio<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O <strong>quadrado de um trin\u00f4mio<\/strong> (polin\u00f4mio formado por 3 termos) \u00e9 igual ao quadrado do primeiro termo, mais o quadrado do segundo termo, mais o quadrado do terceiro termo, mais duas vezes o primeiro pelo segundo, mais duas vezes o primeiro pelo terceiro, mais o dobro do segundo pelo terceiro. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/carre-dun-trinome.png\" alt=\"Quais s\u00e3o as f\u00f3rmulas para todas as identidades, produtos ou igualdades not\u00e1veis?\" class=\"wp-image-2847\" width=\"362\" height=\"290\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> De qualquer trin\u00f4mio ao quadrado:<\/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-379affdd0954c4ca08ed08041e0eb7b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O quadrado acima pode ser fatorado no trin\u00f4mio multiplicado por ele mesmo:<\/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-d8944601270bfee61c23bb9440e7fd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)^2 = (a+b+c)(a+b+c)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"275\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora resolvemos a multiplica\u00e7\u00e3o polinomial:<\/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-ceaddc98a341af8c426098e15affbe7a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)(a+b+c)= a^2+ab+ac+ba+b^2+bc+ca+cb+c^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"505\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E por fim, agrupamos termos semelhantes:<\/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-79d99fd9567501331249064ee77e6db1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2+ab+ac+ba+b^2+bc+ca+cb+c^2 = a^2+b^2+c^2+2ab+2ac+2bc\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"576\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Desta forma, j\u00e1 chegamos \u00e0 express\u00e3o da f\u00f3rmula, ent\u00e3o fica demonstrada a f\u00f3rmula do quadrado de um trin\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-a9597e2a9cf6403902d36e5ca6411045_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"345\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h4 class=\"wp-block-heading\">Exemplo:<\/h4>\n<ul>\n<li> Encontre a seguinte igualdade not\u00e1vel:<\/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-7ee4db6a5192b7efea2342d21275e487_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left(x^2+x+3\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"101\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula do quadrado de um trin\u00f4mio \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-a9597e2a9cf6403902d36e5ca6411045_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"345\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como em todas as igualdades not\u00e1veis, primeiro voc\u00ea deve identificar os valores das inc\u00f3gnitas na f\u00f3rmula. Neste exerc\u00edcio<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> Leste<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-09f6edd3d7af07ab26b4a0a71c20c0b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2,\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\"><\/p>\n<p> o coeficiente<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponder ao<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-038741496726a75b03e91a2e030b0287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o termo independente 3:<\/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-55e06f44486e75e9153a60d36e83bc37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} (a+b+c)^2\\\\[2ex] \\left(x^2+x+3\\right)^2 \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=x^2 \\\\[2ex] b=x \\\\[2ex] c=3 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"90\" width=\"343\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E quando j\u00e1 sabemos os valores, basta substituir esses valores na f\u00f3rmula e fazer os c\u00e1lculos: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-de-trinome-au-carre.png\" alt=\"calculadora de identidades, produtos e igualdades not\u00e1veis\" class=\"wp-image-2850\" width=\"643\" height=\"224\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Identidades-o-productos-notables-al-cubo\"><\/span> Identidades (ou produtos) not\u00e1veis ao cubo<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Acabamos de estudar todas as identidades not\u00e1veis ao quadrado, ou seja, todos os tipos de identidades not\u00e1veis que s\u00e3o formadas por pot\u00eancias elevadas a 2. Bom, agora vamos analisar as identidades not\u00e1veis ao cubo. \u00c9 claro que as f\u00f3rmulas de identidade ao cubo s\u00e3o um pouco mais complicadas, mas tamb\u00e9m s\u00e3o muito \u00fateis.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cubo-de-una-suma\"><\/span> cubo de uma soma<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O not\u00e1vel <strong>produto c\u00fabico de uma soma<\/strong> \u00e9 um bin\u00f4mio (polin\u00f4mio com apenas dois mon\u00f4mios) elevado a 3 cujos dois elementos s\u00e3o positivos. Portanto, algebricamente, o cubo de uma soma \u00e9 expresso como <strong>(a+b) <sup>3<\/sup><\/strong> .<\/p>\n<p> A f\u00f3rmula para a igualdade not\u00e1vel do cubo de uma soma \u00e9: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/binome-dune-somme-ou-somme-au-cube-formule.png\" alt=\"Quais s\u00e3o todos os produtos, identidades ou v\u00ednculos not\u00e1veis?\" class=\"wp-image-2810\" width=\"280\" height=\"280\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Partindo de um bin\u00f4mio positivo ao cubo:<\/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-380239d5f1b18e11f3b6b0931a4f14d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A pot\u00eancia acima pode ser fatorada no produto do fator<\/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-2cfde4aacc08aae1ed70fe5f7b2f74de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> pelo seu quadrado:<\/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-d89c1125bf18f5ec3b34a3bc8e4de45b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^3=(a+b)\\cdot (a+b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Da mesma forma, como vimos em not\u00e1veis igualdades quadradas, o 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-2cfde4aacc08aae1ed70fe5f7b2f74de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> Pode ser resolvido com a f\u00f3rmula do quadrado de uma soma:<\/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-4b1c6920425dd90a9526a1eaccf056b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\\cdot (a+b)^2=(a+b)\\cdot (a^2+2ab+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"328\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o multiplicamos os dois polin\u00f4mios:<\/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-06771ecbb13542eae2a68477f849d729_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a+b)\\cdot (a^2+2ab+b^2) &amp; = a\\cdot a^2 +a\\cdot 2ab + a\\cdot b^2+b\\cdot a^2 +b\\cdot 2ab +b \\cdot b^2 \\\\[2ex] &amp; = a^3+2a^2b+ab^2+ba^2+2ab^2+b^3 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"555\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, s\u00f3 temos que agrupar termos semelhantes:<\/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-27d0da0e0e3ce760508c47f425fd1d68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3+2a^2b+ab^2+ba^2+2ab^2+b^3 = a^3+3a^2b+3ab^2+b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"445\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E assim se verifica a f\u00f3rmula para a identidade not\u00e1vel de uma soma binomial ao cubo: <\/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-536cf8075ed9dc1e16eb5da114b79756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^3 = a^3+3a^2b+3ab^2 +b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Resumindo, uma soma elevada a 3 \u00e9 igual ao cubo do primeiro, mais tr\u00eas vezes o quadrado do primeiro pelo segundo, mais tr\u00eas vezes o primeiro pelo quadrado do segundo, mais o cubo do segundo.<\/p>\n<h4 class=\"wp-block-heading\"> Exemplo:<\/h4>\n<ul>\n<li> Resolva a seguinte identidade not\u00e1vel de uma soma ao cubo usando sua f\u00f3rmula correspondente:<\/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-9b6665bb45814802bf3d7dbb8b68c771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x+2)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"61\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste problema temos um bin\u00f4mio elevado a 3 cujos dois termos s\u00e3o positivos. Devemos, portanto, usar a f\u00f3rmula para uma soma ao cubo:<\/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-536cf8075ed9dc1e16eb5da114b79756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^3 = a^3+3a^2b+3ab^2 +b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agora precisamos encontrar o valor dos par\u00e2metros<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula. Nesse caso,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> corresponde \u00e0 vari\u00e1vel<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o n\u00famero 2.<\/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-909b3b4a2f976c165f160a6765b3ed9d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} (a+b)^3\\\\[2ex] (x+2)^3 \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=x \\\\[2ex] b=2 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"295\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Com o qual calculamos o produto not\u00e1vel substituindo os valores de<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> na f\u00f3rmula: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/exemple-dun-binome-somme-et-difference-au-cube.jpg\" alt=\"10 exemplos de produtos ou identidades not\u00e1veis\" class=\"wp-image-2468\" width=\"419\" height=\"168\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Cubo-de-una-diferencia\"><\/span> cubo de uma diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> O <strong>cubo de uma diferen\u00e7a<\/strong> , ou <strong>cubo de uma subtra\u00e7\u00e3o<\/strong> , \u00e9 um bin\u00f4mio elevado a 3 que possui um termo com sinal negativo. Portanto, a express\u00e3o matem\u00e1tica para este not\u00e1vel tipo de produto \u00e9 <strong>(ab) <sup>3<\/sup><\/strong> .<\/p>\n<p> A f\u00f3rmula para o cubo de uma diferen\u00e7a (ou subtra\u00e7\u00e3o) \u00e9: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/binome-dune-soustraction-de-difference-de-la-formule-du-cube.png\" alt=\"identidades, produtos ou igualdades c\u00fabicas not\u00e1veis\" class=\"wp-image-2811\" width=\"279\" height=\"279\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\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 a demonstra\u00e7\u00e3o da f\u00f3rmula<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Obviamente, a prova desta f\u00f3rmula \u00e9 muito semelhante \u00e0 do produto not\u00e1vel de uma soma ao cubo. Mas neste caso, partimos de um bin\u00f4mio c\u00fabico negativo:<\/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-6a48c875098bfee5d50068c1f0e7296d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"59\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Claramente, a potencia\u00e7\u00e3o anterior pode ser decomposta no produto do fator<\/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-5ab5e2adaf0a63382c066ea55b51147c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> multiplicado pelo seu quadrado:<\/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-ada6004c3907e554bc5bde167ff16a0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^3=(a-b)\\cdot (a-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, como estudamos em identidades quadradas not\u00e1veis, o 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-5ab5e2adaf0a63382c066ea55b51147c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"51\" style=\"vertical-align: -5px;\"><\/p>\n<p> Pode ser calculado com a f\u00f3rmula do quadrado da diferen\u00e7a:<\/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-c2570ca2db47f67aa0eaf670615e2743_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)\\cdot (a-b)^2=(a-b)\\cdot (a^2-2ab+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"328\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora produzimos o produto dos dois polin\u00f4mios:<\/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-627a273de8fff974f4a14a32fcee90b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (a-b)\\cdot (a^2-2ab+b^2) &amp; = a\\cdot a^2 +a\\cdot (-2ab) + a\\cdot b^2-b\\cdot a^2 -b\\cdot (-2ab)-b \\cdot b^2 \\\\[2ex] &amp; = a^3-2a^2b+ab^2-ba^2+2ab^2-b^3 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"610\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E o \u00faltimo passo \u00e9 agrupar termos semelhantes:<\/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-62f63e77f52ddb89cdd2e650938edb82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3-2a^2b+ab^2-ba^2+2ab^2-b^3 = a^3-3a^2b+3ab^2-b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"445\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim se verifica a f\u00f3rmula da identidade not\u00e1vel de um bin\u00f4mio subtra\u00eddo elevado ao cubo: <\/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-5a9a96bd2d1f115178fbbcf19c8047c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^3 = a^3-3a^2b+3ab^2-b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Portanto, uma diferen\u00e7a (ou subtra\u00e7\u00e3o) elevada a tr\u00eas \u00e9 igual ao cubo do primeiro, menos tr\u00eas vezes o quadrado do primeiro pelo segundo, mais tr\u00eas vezes o primeiro pelo quadrado do segundo, menos o cubo do segundo.<\/p>\n<h4 class=\"wp-block-heading\"> Exemplo:<\/h4>\n<ul>\n<li> Calcule o pr\u00f3ximo bin\u00f4mio ao cubo (diferen\u00e7a) usando sua f\u00f3rmula correspondente:<\/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-fb0dbc34da8cea6a7d6622c9a3c5faba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3x-2)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste exerc\u00edcio, temos um par com um elemento positivo e um elemento negativo. Devemos, portanto, usar a f\u00f3rmula para uma diferen\u00e7a ao cubo:<\/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-746a96ec30fac619eedf62054c377fe5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^3 = a^3-3a^2b+3ab^2 -b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Primeiro, como sempre, identificamos o valor das inc\u00f3gnitas<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> da f\u00f3rmula. Nesse caso<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> representa o mon\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bcea841b93e6d1c6150bf94b4036ab3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o termo independente do bin\u00f4mio, ou seja, 2.<\/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-a792ec6dead8466ec6a2cb2a43d9fab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{l} (a-b)^3\\\\[2ex] (3x-2)^3 \\end{array} \\color{red} \\right\\} \\quad \\color{red}\\bm{\\longrightarrow}\\quad  \\color{black} \\begin{array}{c} a=3x \\\\[2ex] b=2 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Observe que o par\u00e2metro<\/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-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 simplesmente igual a 2, sem o sinal negativo do n\u00famero. \u00c9 importante ter isso em mente para aplicar corretamente a f\u00f3rmula.<\/p>\n<p> Finalmente, encontramos a identidade not\u00e1vel colocando os valores de<\/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-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> e de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> na f\u00f3rmula: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/binome-cube-negatif-parfait.jpg\" alt=\"desenvolver identidades, produtos e igualdades not\u00e1veis\" class=\"wp-image-2476\" width=\"501\" height=\"169\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Tabla-resumen-de-las-identidades-notables\"><\/span> Tabela resumo de identidades not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Em resumo, fizemos uma tabela com todas as identidades (ou produtos) not\u00e1veis que vimos, assim ser\u00e1 mais f\u00e1cil para voc\u00ea estud\u00e1-las. \ud83d\ude09 <\/p>\n<figure class=\"wp-block-image size-large\"><img decoding=\"async\" loading=\"lazy\" width=\"1024\" height=\"525\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formules-des-identites-produits-ou-egalites-remarquables.png\" alt=\"f\u00f3rmulas de identidades ou igualdades de produtos not\u00e1veis\" class=\"wp-image-2808\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-identidades-o-productos-notables\"><\/span> Exerc\u00edcios resolvidos de identidades (ou produtos) not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para que voc\u00ea termine de entender a no\u00e7\u00e3o de identidades not\u00e1veis, tamb\u00e9m chamadas de produtos not\u00e1veis ou igualdades not\u00e1veis, preparamos diversos exerc\u00edcios resolvidos passo a passo. Voc\u00ea pode tentar faz\u00ea-los e depois verificar se se saiu bem com as solu\u00e7\u00f5es dos exerc\u00edcios.<\/p>\n<p class=\"has-text-align-center\"> \u2b07\u2b07 N\u00e3o esque\u00e7a que voc\u00ea pode nos tirar todas as suas d\u00favidas abaixo nos coment\u00e1rios! \u2b07\u2b07<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Expanda as seguintes identidades not\u00e1veis (soma de quadrados): <\/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-9229e4ae2034182594cea6b72883a61e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ (x+3)^2\" 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-837ba9382325be793705fe7f068579be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ (6x+2)^2\" 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-55d230fdf4d87dfb39f5427089c4bcd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(x^2+7\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -7px;\"><\/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-8858ec9b4957c47679e682ab433bd75d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ (5x+8y)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"106\" 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-left\"> Todas as identidades not\u00e1veis no problema s\u00e3o somas quadradas, portanto neste caso devemos aplicar sempre a mesma 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-e3c7bb69fbb939444db4e075615462f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2=a^2+2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" 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-571dada676a093b9b625887a09615b5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}(x+3)^2&amp; =x^2+2\\cdot x\\cdot 3 +3^2\\\\[2ex] &amp; = \\bm{x^2+6x +9}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"248\" 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-067fdf38612ca481db587bda479cab24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}(6x+2)^2 &amp; =(6x)^2+2\\cdot 6x \\cdot 2+2^2\\\\[2ex] &amp; = \\bm{36x^2+24x+4}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"288\" 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-62f7ef68fc47d45958f6a10dbfe3f512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned}\\left(x^2+7\\right)^2 &amp; = \\left(x^2\\right)^2+2\\cdot x^2\\cdot 7 +7^2\\\\[2ex] &amp; = \\bm{x^4+14x^2 +49}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"290\" 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-2fdf798e7d585cdbc2bbeb0417bfc62a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}(5x+8y)^2 &amp; =(5x)^2+2\\cdot 5x\\cdot 8y +(8y)^2\\\\[2ex] &amp; = \\bm{25x^2+80xy+64y^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"331\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 2<\/h3>\n<p> Desenvolva os seguintes produtos not\u00e1veis (diferen\u00e7as ao quadrado): <\/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-4c11a4f53553874acb14ec9bbd0c78d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ (x-2)^2\" 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-55330f15a13171e004a6fd9063b5042d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ (3-7x)^2\" 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-9a9afedffeffbe27f4e9c5d94b2bcad2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(x^2-6\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -7px;\"><\/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-f96b2af0d8ddde63f0f5ff04acde9e8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ (-3x+y)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"111\" 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-left\"> Todos os produtos not\u00e1veis neste exerc\u00edcio s\u00e3o subtra\u00e7\u00f5es ao quadrado, portanto s\u00f3 precisamos aplicar uma 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-3074d6e8bc69734f38234657d1fddc4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^2=a^2-2ab+b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" 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-14d502eda968fe82617b4403cd9c4722_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}(x-2)^2&amp; =x^2-2\\cdot x\\cdot 2 +2^2\\\\[2ex] &amp; = \\bm{x^2-4x +4}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"248\" 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-c22d520301280872e645f5683a2fba8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}(3-7x)^2 &amp; =3^2-2\\cdot 3\\cdot 7x +(7x)^2\\\\[2ex] &amp; = \\bm{9-42x+49x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"288\" 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-95c7c481a96b20b700bd2253c90f0c0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned}\\left(x^2-6\\right)^2 &amp; = \\left(x^2\\right)^2-2\\cdot x^2\\cdot 6 +6^2\\\\[2ex] &amp; = \\bm{x^4-12x^2 +36}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"290\" 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-3cea9fa89580d3d9d9df7fd93cca2b89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}(-3x+y)^2 &amp; = (y-3x)^2 \\\\[2ex] &amp; = y^2-2\\cdot y\\cdot 3x +(3x)^2\\\\[2ex] &amp; = \\bm{y^2-6yx+9x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"109\" width=\"304\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 3<\/h3>\n<p> Desenvolva as seguintes igualdades not\u00e1veis (produtos de somas por diferen\u00e7as): <\/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-3a101dc0bb3e2ab901e5a441cdb22369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ (x+5)(x-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" 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-5a3cc5a2bfc3829e05fbf0cc6fd4dea9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ (2x+6)(2x-6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"151\" 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-5b3b43f8e1142337f367f44c27632578_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ (x+7)(x-7)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" 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-80db9b8cb45a16f2c4b4dd810f0ef940_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ (x-4y)(x+4y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"153\" 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-left\"> Como todas as igualdades not\u00e1veis neste exerc\u00edcio s\u00e3o multiplica\u00e7\u00f5es de somas por diferen\u00e7as, todas elas s\u00e3o resolvidas com a mesma 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-e1868f84409086d4b0b21464e4a4f207_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)\\cdot (a-b) =a^2-b^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"194\" 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-826c4aec8f005514a14cdc8555c084c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}(x+5)(x-5) &amp;=x^2-5^2\\\\[2ex] &amp; = \\bm{x^2-25}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"221\" 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-6793239af84413fb9408c2cb6033e5ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}(2x+6)(2x-6) &amp; =(2x)^2-6^2 \\\\[2ex] &amp; = \\bm{4x^2-36}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"260\" 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-630b94cf4be27c5f7b9c87651368634d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned}(x+7)(x-7) &amp; =x^2-7^2 \\\\[2ex] &amp; = \\bm{x^2-49}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"61\" width=\"221\" 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-80c5451e407a2c0e670c6cb22a74043c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}(x-4y)(x+4y) &amp; =(x+4y)(x-4y) \\\\[2ex] &amp; =x^2-(4y)^2\\\\[2ex] &amp; = \\bm{x^2-16y^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"106\" width=\"310\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 4<\/h3>\n<p> Resolva todas as seguintes identidades not\u00e1veis: <\/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-f11abde1a676c9efe0dee6544ec7dd35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\left(x^2+10\\right)\\left(x^2-10\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"177\" style=\"vertical-align: -7px;\"><\/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-d91f0bc70f84a13bc544db52068d89d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(4x^2+2y^3\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"126\" style=\"vertical-align: -7px;\"><\/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-e13ca38c8d8b96d56e5c72d75ab1db90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(6x^3-4y^4\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"126\" style=\"vertical-align: -7px;\"><\/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-33d63537ffb97df07d85a50a5bd46561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\left(8x^3+y^2\\right)\\left(8x^3-y^2\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"193\" style=\"vertical-align: -7px;\"><\/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-0d4df947d503beab8b2af90de2d8d605_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E)} \\ \\left(5x^2-9x\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"118\" style=\"vertical-align: -7px;\"><\/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-c65875e01d82840e30ae85d803d45e90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}\\left(x^2+10\\right)\\left(x^2-10\\right) &amp; =\\left(x^2\\right)^2-10^2\\\\[2ex] &amp; = \\bm{x^4-100}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"64\" width=\"294\" 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-04e0bcf5df362d320cfdb2f87cdc6ddc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}\\left(4x^2+2y^3\\right)^2 &amp; =\\left(4x^2\\right)^2+2\\cdot 4x^2\\cdot 2y^3 +\\left(2y^3\\right)^2\\\\[2ex] &amp; = \\bm{16x^4+16x^2y^3+4y^6}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"383\" 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-cc3f7dc61f7c44a60c01e0a95de278fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned}\\left(6x^3-4y^4\\right)^2 &amp;  =\\left(6x^3\\right)^2-2\\cdot 6x^3\\cdot 4y^4 +\\left(4y^4\\right)^2 = \\\\[2ex] &amp;= \\bm{36x^6-48x^3y^4+16y^8}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"402\" 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-a4d4a0c86d26820881eb65cb92c3679a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}\\left(8x^3+y^2\\right)\\left(8x^3-y^2\\right) &amp; =\\left(8x^3\\right)^2-\\left(y^2\\right)^2 \\\\[2ex] &amp; = \\bm{64x^6-y^4}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"335\" 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-432c4ae0f050bec15e3fa52f426698ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{E)} \\ \\begin{aligned}\\left(5x^2-9x\\right)^2 &amp; =\\left(5x^2\\right)^2-2\\cdot 5x^2\\cdot 9x +\\left(9x\\right)^2 \\\\[2ex] &amp; = \\bm{25x^4-90x^3+81x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"66\" width=\"360\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 5<\/h3>\n<p> Calcule os seguintes produtos not\u00e1veis: <\/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-ca75501df4056045f750323893bee27c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ (x+4)^3\" 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-f1ae5aad30adde91e86d6bb696b6adc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(x^2-5\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -7px;\"><\/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-debc2fbed1f43204a1d3b191ce697175_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(2x-1\\right)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"99\" 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-641ad931932e2d32742c712339a76903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ (5x+2)^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"97\" 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-left\"> Para encontrar todos os produtos not\u00e1veis do problema \u00e9 necess\u00e1rio aplicar as f\u00f3rmulas de soma e diferen\u00e7a ao cubo conforme os casos: <\/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-536cf8075ed9dc1e16eb5da114b79756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^3 = a^3+3a^2b+3ab^2 +b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" 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-5a9a96bd2d1f115178fbbcf19c8047c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a-b)^3 = a^3-3a^2b+3ab^2-b^3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"247\" 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-14695fb807e2df89352fdd1c1dced2ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}(x+4)^3&amp; =x^3+3\\cdot x^2\\cdot 4 +3\\cdot x\\cdot 4^2+4^3\\\\[2ex] &amp; =x^3+3\\cdot x^2\\cdot 4 +3\\cdot x\\cdot 16+64 \\\\[2ex] &amp; = \\bm{x^3+12x^2+48x+64}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"342\" 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-5be0d584351feb0bef5572ca5c9e159a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}\\left(x^2-5\\right)^3&amp; =\\left(x^2\\right)^3-3\\cdot \\left(x^2\\right)^2\\cdot 5 +3\\cdot x^2\\cdot 5^2-5^3\\\\[2ex] &amp; =x^6-3\\cdot x^4\\cdot 5 +3\\cdot x^2\\cdot 25-125 \\\\[2ex] &amp; = \\bm{x^6-15x^4+75x^2-125}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"110\" width=\"404\" 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-f44f9c3283dad97321644c6e559f64ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{aligned}\\left(2x-1\\right)^3&amp; =\\left(2x\\right)^3-3\\cdot \\left(2x\\right)^2\\cdot 1 +3\\cdot 2x\\cdot 1^2-1^3\\\\[2ex] &amp; =8x^3-3\\cdot 4x^2\\cdot 1 +3\\cdot 2x\\cdot 1-1 \\\\[2ex] &amp; = \\bm{8x^3-12x^2+6x-1}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"401\" 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-156e7619e4d6ef129f04250af8197d2e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}(5x+2)^3&amp; =(5x)^3+3\\cdot \\left(5x\\right)^2\\cdot 2 +3\\cdot 5x\\cdot 2^2+2^3\\\\[2ex] &amp; =125x^3+3\\cdot 25x^2\\cdot 2 +3\\cdot 5x\\cdot 4+8 \\\\[2ex] &amp; = \\bm{125x^3+150x^2+60x+8}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"402\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 6<\/h3>\n<p> Resolva as seguintes igualdades not\u00e1veis: <\/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-b1e2db1dacfd70bcdf33589968427633_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\left(x^2+x+5\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"132\" style=\"vertical-align: -7px;\"><\/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-6eddfbe2255cd7f5b8d829ff6aef4e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(x^2+3x-4\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"141\" style=\"vertical-align: -7px;\"><\/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-577a572f8974257e1d6d7b8411f75f4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(4x^2-6x+3\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"150\" style=\"vertical-align: -7px;\"><\/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-4a26869f17cf7889324d3d5e6755b800_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\left(x^3-3x^2-9x\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"159\" style=\"vertical-align: -7px;\"><\/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-left\"> Para resolver todas essas identidades not\u00e1veis, precisamos usar a f\u00f3rmula do quadrado de um trin\u00f4mio, que \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-a9597e2a9cf6403902d36e5ca6411045_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b+c)^2=a^2+b^2+c^2+2ab+2ac+2bc\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"345\" 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-749dc45e7a00d7122d62b774706bdcc0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l} \\left(x^2+x+5\\right)^2 = \\\\[2ex] = \\left(x^2\\right)^2+x^2+5^2+2\\cdot x^2 \\cdot x + 2 \\cdot x^2 \\cdot 5 +2 \\cdot x \\cdot 5 = \\\\[2ex] = x^4+x^2+25+2x^3 + 10x^2 +10x = \\\\[2ex] = \\bm{x^4+2x^3+11x^2+10x+25} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"438\" 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-b1f51f18b3c1118b6e8e3acc3441b0ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l}\\left(x^2+3x-4\\right)^2 = \\\\[2ex] = \\left(x^2\\right)^2+(3x)^2+(-4)^2+2\\cdot x^2 \\cdot 3x + 2 \\cdot x^2 \\cdot (-4) +2 \\cdot 3x \\cdot (-4) = \\\\[2ex] = x^4+9x^2+16+6x^3-8x^2-24x = \\\\[2ex] = \\bm{x^4+6x^3+x^2-24x+16} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"557\" 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-49c6496bf684296d315fc96d9cb5857e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l}\\left(4x^2-6x+3\\right)^2 = \\\\[2ex] = \\left(4x^2\\right)^2+(-6x)^2+3^2+2\\cdot 4x^2 \\cdot (-6x) + 2 \\cdot 4x^2 \\cdot 3 +2 \\cdot (-6x) \\cdot 3 = \\\\[2ex] = 16x^4+36x^2+9-48x^3+24x^2-36x = \\\\[2ex] = \\bm{16x^4-48x^3+60x^2-36x+9} \\end{array}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"570\" 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-7cd08035d8402c27c411bcf5b30216cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{array}{l}  \\left(x^3-3x^2-9x\\right)^2 = \\\\[2ex] = \\left(x^3\\right)^2+\\left(-3x^2\\right)^2+(-9x)^2+2\\cdot x^3 \\cdot (-3x^2) + 2 \\cdot x^3 \\cdot (-9x) +2 \\cdot (-3x^2) \\cdot (-9x) = \\\\[2ex] = x^6+9x^4+81x^2-6x^5-18x^4+54x^3 = \\\\[2ex] = \\bm{x^6-6x^5-9x^4+54x^3+81x^2} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"136\" width=\"682\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 7<\/h3>\n<p> Calcule as seguintes identidades not\u00e1veis com ra\u00edzes e fra\u00e7\u00f5es (alta dificuldade): <\/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-2fa707460c7ffd54eac2fb73d35c6734_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\displaystyle \\left(\\sqrt{2x}-\\sqrt{8x}\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"146\" style=\"vertical-align: -11px;\"><\/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-3bfc1a9172f021d35179b9df54d8a126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\displaystyle \\left(\\frac{1}{2}x^2+\\frac{5}{3}x\\right)\\left(\\frac{1}{2}x^2-\\frac{5}{3}x\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"231\" style=\"vertical-align: -17px;\"><\/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-b432940794af539924a002fac6533134_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\displaystyle \\left(\\frac{4}{3}x^2+\\frac{3}{2}x\\right)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"137\" style=\"vertical-align: -17px;\"><\/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-47d530888455eb1be7950e4d42776002_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\Bigl(9x^3+\\sqrt{5x}\\Bigr)\\Bigl(9x^3-\\sqrt{5x}\\Bigr)\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"230\" style=\"vertical-align: -11px;\"><\/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-left\"> A se\u00e7\u00e3o A) consiste em uma subtra\u00e7\u00e3o ao quadrado, portanto para resolv\u00ea-la deve-se aplicar sua f\u00f3rmula correspondente e, al\u00e9m disso, deve-se lembrar que se uma raiz for quadrada ela \u00e9 simplificada:<\/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-999e71bf062ea313780439abaf2b4295_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{aligned}\\left(\\sqrt{2x}-\\sqrt{8x}\\right)^2 &amp; =\\left(\\sqrt{2x}\\right)^2-2\\cdot \\sqrt{2x}\\cdot \\sqrt{8x} +\\left(\\sqrt{8x}\\right)^2\\\\[2ex] &amp; =2x-2\\sqrt{2x\\cdot 8x} +8x \\\\[2ex] &amp; = 10x-2\\sqrt{16x^2} \\\\[2ex] &amp;= 10x-2\\cdot 4x = \\\\[2ex] &amp; = 10x -8x \\\\[2ex] &amp; = \\bm{2x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"247\" width=\"444\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A se\u00e7\u00e3o B) trata da adi\u00e7\u00e3o por subtra\u00e7\u00e3o e os mon\u00f4mios possuem coeficientes fracion\u00e1rios, com os quais este produto not\u00e1vel deve ser determinado usando a f\u00f3rmula de adi\u00e7\u00e3o por subtra\u00e7\u00e3o e as propriedades das fra\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-24593bac7bd4a9837e1f18fef4f9c38e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{aligned}\\displaystyle \\left(\\frac{1}{2}x^2+\\frac{5}{3}x\\right)\\left(\\frac{1}{2}x^2-\\frac{5}{3}x\\right) &amp; \\displaystyle =\\left(\\frac{1}{2}x^2\\right)^2-\\left(\\frac{5}{3}x\\right)^2\\\\[4ex] \\displaystyle &amp; =\\frac{1^2}{2^2}x^4-\\frac{5^2}{3^2}x^2\\\\[4ex]\\displaystyle &amp; = \\mathbf{\\frac{1}{4}}\\bm{x^4-}\\mathbf{\\frac{25}{9}}\\bm{x^2} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"195\" width=\"402\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A not\u00e1vel igualdade na se\u00e7\u00e3o C) \u00e9 uma soma elevada a 2 e, da mesma forma, \u00e9 composta de fra\u00e7\u00f5es. Portanto, para calcul\u00e1-lo precisamos usar a f\u00f3rmula da soma quadrada mais as propriedades das fra\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-c50dcca740e334b34f746e71f4af826e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\displaystyle \\begin{aligned} \\left(\\frac{4}{3}x^2+\\frac{3}{2}x\\right)^2 &amp; = \\left(\\frac{4}{3}x^2\\right)^2+2\\cdot \\frac{4}{3}x^2\\cdot \\frac{3}{2}x +\\left(\\frac{3}{2}x\\right)^2\\\\[2ex] &amp; = \\frac{4^2}{3^2}x^4+2\\cdot \\frac{12}{6}x^3 +\\frac{3^2}{2^2}x^2 \\\\[2ex] &amp;= \\frac{16}{9}x^4 +2\\cdot 2x^3+\\frac{9}{4}x^2 \\\\[2ex] &amp; = \\mathbf{\\frac{16}{9}} \\bm{x^4+4x^3+}\\mathbf{\\frac{9}{4}}\\bm{x^2}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"222\" width=\"416\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A \u00faltima identidade not\u00e1vel trata de uma soma vezes uma diferen\u00e7a com coeficientes irracionais, ent\u00e3o aplicamos a f\u00f3rmula para uma soma vezes uma diferen\u00e7a e depois simplificamos as ra\u00edzes quadradas:<\/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-7c540e4315e9e84faaa2ff656c4eec21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{aligned}\\Bigl(9x^3+\\sqrt{5x}\\Bigr)\\Bigl(9x^3-\\sqrt{5x}\\Bigr) &amp; =\\Bigl(9x^3\\Bigr)^2-\\left(\\sqrt{5x}\\right)^2\\\\[2ex] &amp; = \\bm{81x^6-5x}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"73\" width=\"402\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Otros-tipos-de-identidades-notables\"><\/span> Outros tipos de identidade not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Todas as identidades not\u00e1veis que discutimos acima s\u00e3o as mais comumente usadas. No entanto, em matem\u00e1tica existem outros tipos de produtos not\u00e1veis que tamb\u00e9m s\u00e3o interessantes de conhecer, pois s\u00e3o utilizados para diversos fins.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Suma-de-cubos\"><\/span> soma de cubos<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>soma dos cubos<\/strong> corresponde a um bin\u00f4mio cujos dois termos s\u00e3o positivos e, al\u00e9m disso, suas ra\u00edzes c\u00fabicas s\u00e3o exatas. Portanto, a express\u00e3o alg\u00e9brica para uma soma de cubos \u00e9 <strong>a <sup>3<\/sup> +b <sup>3<\/sup><\/strong> . <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-la-somme-des-cubes.png\" alt=\"identidades, produtos ou v\u00ednculos not\u00e1veis resolvidos\" class=\"wp-image-2663\" width=\"306\" height=\"307\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A f\u00f3rmula deste produto not\u00e1vel \u00e9 utilizada para fatorar um polin\u00f4mio, ou seja, atrav\u00e9s da f\u00f3rmula transformamos um polin\u00f4mio em produto de um bin\u00f4mio por um trin\u00f4mio.<\/p>\n<p> Para que voc\u00ea possa ver como isso \u00e9 feito, aqui est\u00e1 um exemplo de aplica\u00e7\u00e3o desta not\u00e1vel identidade:<\/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-51a33d4fd52d94afa78abd4be81cf7f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3+8\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"48\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Na verdade, a express\u00e3o anterior consiste numa adi\u00e7\u00e3o de cubos porque a raiz c\u00fabica do mon\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-a5e0e31e823b4d5c9a90c0d01d5e8fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 exato (n\u00e3o fornece n\u00famero decimal) e o n\u00famero 8 tamb\u00e9m: <\/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-1105a3d4349d8c5d3eae7b16dc079ef1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt[3]{x^3} = x\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -1px;\"><\/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-71ce4de717d54a2fb6c3282de038913a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt[3]{8} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"54\" style=\"vertical-align: -2px;\"><\/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-4171629bd68508074adfbf81cf982b5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3+8=x^3+2^3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"127\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Portanto, podemos usar a f\u00f3rmula da soma de cubos perfeitos para transformar a express\u00e3o c\u00fabica em produto de um bin\u00f4mio por um trin\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-f673c682dcdc4e38ce08e8a77cf4e7f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3+b^3 = (a+b)(a^2-ab+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"236\" 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-f30ea5f0f7ef1b89a16f1d00e54d063c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} x^3 +2^3 &amp; = (x+2)(x^2-x \\cdot 2 + 2^2) \\\\[2ex] &amp; = (x+2)(x^2-2x + 4) \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"256\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Diferencia-de-cubos\"><\/span>diferen\u00e7a de cubos<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>diferen\u00e7a (ou subtra\u00e7\u00e3o) de cubos<\/strong> \u00e9 um bin\u00f4mio composto por um termo positivo e um termo negativo cujas ra\u00edzes c\u00fabicas s\u00e3o exatas. Em outras palavras, uma diferen\u00e7a de cubos \u00e9 expressa na forma <strong>a <sup>3<\/sup> -b <sup>3<\/sup><\/strong> . <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-pour-la-difference-ou-la-soustraction-de-cubes.png\" alt=\"Exerc\u00edcios resolvidos para fatorar polin\u00f4mios com identidades not\u00e1veis\" class=\"wp-image-2731\" width=\"305\" height=\"306\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Vamos dar um exemplo para voc\u00ea ver como esse not\u00e1vel tipo de identidade \u00e9 resolvido:<\/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-3cb03de00c11c61a46f0473cac25b903_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3-27\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> \u00c9 uma diferen\u00e7a de cubos porque tanto a raiz c\u00fabica do mon\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-a5e0e31e823b4d5c9a90c0d01d5e8fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\"><\/p>\n<p> como 27 est\u00e3o corretos: <\/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-1105a3d4349d8c5d3eae7b16dc079ef1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt[3]{x^3} = x\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -1px;\"><\/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-4379f1587711ba1048df5a84748d12da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\sqrt[3]{27} = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"64\" style=\"vertical-align: -2px;\"><\/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-64a522b09529e310087510320b8c3ad6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^3-27=x^3-3^3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Voc\u00ea pode, portanto, usar a f\u00f3rmula da diferen\u00e7a de cubos perfeitos para fatorar o 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-52d5ddbcfea3f7d3d492b8f0ead32dc2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3-b^3  = (a-b)(a^2+ab+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"236\" 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-342a448f849bf2856ad9a5394733faeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} x^3 -3^3 &amp; = (x-3)(x^2+x \\cdot 3 + 3^2) \\\\[2ex] &amp; =(x-3)(x^2+3x + 9)  \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"256\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Producto-de-binomios-con-un-termino-comun\"><\/span> Produto de bin\u00f4mios com um termo comum<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Este produto not\u00e1vel \u00e9 usado para converter um produto de dois bin\u00f4mios que possuem um termo comum em um polin\u00f4mio quadr\u00e1tico. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/formule-de-deux-binomes-avec-un-terme-en-commun-2.png\" alt=\"identidades, produtos ou igualdades not\u00e1veis pdf\" class=\"wp-image-2793\" width=\"264\" height=\"265\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Aqui est\u00e1 um exemplo elaborado deste tipo de produto not\u00e1vel: <\/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-8447db6a2246c09b2e7be29f8050a3d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} (x+4)(x+5) &amp;= x^2+(4+5)x+4\\cdot 5 \\\\[2ex] &amp; = x^2+9x+20 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"62\" width=\"286\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Mas-identidades\"><\/span> mais identidades<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Embora as identidades not\u00e1veis sejam as mais famosas por serem as mais comuns, deve-se notar que tamb\u00e9m existem mais identidades com outros nomes. Aqui est\u00e1 uma lista de outras identidades menos conhecidas, caso voc\u00ea esteja curioso:<\/p>\n<ul style=\"color:#ff5733; font-weight: bold;\">\n<li style=\"margin-bottom:10px\"> <span style=\"color:#000000;font-weight: normal;\">Identidades de Lagrange:<\/span> <\/li>\n<ul style=\"list-style-type:circle\">\n<li style=\"margin-bottom:10px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a9d12f9a33e8194fbe48dde93ca8918_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a^2+b^2)\\cdot (x^2+y^2) =(ax+by)^2+(ay-bx)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"354\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-439c16d816ff61c2ac4a3c73f04b9a5c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a^2-b^2)\\cdot (x^2-y^2) =(ax+by)^2-(ay+bx)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"354\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<li style=\"margin-bottom:10px\"> <span style=\"color:#000000;font-weight: normal;\">Identidades Legendre:<\/span> <\/li>\n<ul style=\"list-style-type:circle\">\n<li style=\"margin-bottom:10px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-fb371e0857e9b183bb1db9c370d7b779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2+(a-b)^2=2(a^2+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"240\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li style=\"margin-bottom:10px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9b82bb4daacd22465587454eb1ec9350_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^2-(a-b)^2=4ab\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"191\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cc07e405f725019198b41ee5054a97af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(a+b)^4-(a-b)^4=8ab(a^2+b^2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"257\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<li style=\"margin-bottom:10px\"> <span style=\"color:#000000;font-weight: normal;\">Identidade de Argand:<\/span> <\/li>\n<ul style=\"list-style-type:circle\">\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6c8de9c0bcd37989daee33145b0d84cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x^2+x+1)(x^2-x+1) = x^4+x^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"297\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<li style=\"margin-bottom:10px\"> <span style=\"color:#000000;font-weight: normal;\">Identidades gaussianas:<\/span> <\/li>\n<ul style=\"list-style-type:circle\">\n<li style=\"margin-bottom:10px\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d7ad1a682cc650b01984e4a1d9ec2774_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3+b^3+c^3-3abc= (a+b+c)(a^2+b^2+c^2-ab-bc-ac)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"470\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d6b2da7d99ade85355a54bee45b79a9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^3+b^3+c^3-3abc= \\frac{1}{2} (a+b+c)\\left[(a-b)^2+(b-c)^2+(a-c)^2\\right]\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"499\" style=\"vertical-align: -7px;\"><\/p>\n<\/li>\n<\/ul>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Aplicaciones-de-las-identidades-notables\"><\/span> Aplicativos de identidade not\u00e1veis<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Se voc\u00ea chegou at\u00e9 aqui, significa que j\u00e1 sabe fazer c\u00e1lculos com identidades not\u00e1veis. Brilhante! Mas s\u00e9rio\u2026 para que servem as identidades not\u00e1veis? E quando s\u00e3o usadas identidades not\u00e1veis?<\/p>\n<p> Como vimos ao longo deste artigo, o principal objetivo das identidades not\u00e1veis \u00e9 simplificar os c\u00e1lculos. Isto quer dizer que gra\u00e7as a produtos not\u00e1veis podemos resolver diretamente certas pot\u00eancias de polin\u00f4mios complexos sem ter que realizar opera\u00e7\u00f5es dif\u00edceis.<\/p>\n<p> Mas igualdades not\u00e1veis tamb\u00e9m t\u00eam outras fun\u00e7\u00f5es, como fatorar polin\u00f4mios e completar quadrados. A seguir veremos em que consiste cada uma dessas aplica\u00e7\u00f5es. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Factorizacion-de-polinomios\"><\/span> Fatora\u00e7\u00e3o de polin\u00f4mios<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Alguns tipos muito espec\u00edficos de polin\u00f4mios podem ser fatorados com identidades not\u00e1veis. Por exemplo, se encontrarmos um polin\u00f4mio composto por dois termos que s\u00e3o quadrados perfeitos (suas ra\u00edzes quadradas s\u00e3o exatas), podemos fator\u00e1-lo usando a not\u00e1vel f\u00f3rmula de igualdade do produto de uma soma por uma diferen\u00e7a:<\/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-6d6db5dc6ec48fed829d1d16b8803df3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-b^2 =(a+b)(a-b)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"182\" 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-45661a0693691876fa89055734b67833_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-9 =(x+3)(x-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"180\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Da mesma forma, trin\u00f4mios que respeitam as identidades not\u00e1veis do quadrado de uma adi\u00e7\u00e3o ou subtra\u00e7\u00e3o podem ser fatorados: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-3\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-993d882ed2bfdc18bb18dde412cbf270_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2+2ab+b^2=(a+b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" 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-de28c721fc9e8b036c8ed290c4873cf7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2+4x+4=(x+2)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"174\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9cb72c2445ecf74dd260a35c83c91156_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a^2-2ab+b^2=(a-b)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" 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-666a4ad12f00ead4f6e6a1135c228fa2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x^2-10x+25=(x-5)^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"192\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p> Da mesma forma, uma vez fatorado um polin\u00f4mio, as ra\u00edzes (ou zeros) desse polin\u00f4mio podem ser encontradas. Mesmo assim, esse conceito \u00e9 um pouco mais complicado de entender, ent\u00e3o se voc\u00ea tiver mais interesse, recomendamos pesquisar a explica\u00e7\u00e3o no mecanismo de busca do nosso site (canto superior direito), pois temos um artigo completo explicando isso.<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Completacion-de-cuadrados\"><\/span>conclus\u00e3o quadrada<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Completar quadrados \u00e9 um procedimento matem\u00e1tico usado para converter um trin\u00f4mio quadr\u00e1tico na soma de um quadrado mais (ou menos) um n\u00famero.<\/p>\n<p> Dado qualquer trin\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-f7dfafe787a9309542e1e1063e6056ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"ax^2+bx+c\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o o trin\u00f4mio pode ser transformado na seguinte express\u00e3o:<\/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-8304611362475f8451df85e99c1f7675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a(x+h)^2+k\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"104\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> onde os par\u00e2metros<\/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-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> s\u00e3o calculados com as seguintes f\u00f3rmulas:<\/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-6475634f6ee5ca2a7e85945265a0b943_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h=\\cfrac{b}{2a} \\qquad \\qquad k=c-ah^2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"214\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Mesmo que n\u00e3o lhe pare\u00e7a, essas duas f\u00f3rmulas s\u00e3o deduzidas de identidades not\u00e1veis. Assim, gra\u00e7as aos produtos not\u00e1veis, os quadrados podem ser completados.<\/p>\n<p> Como exemplo, aplicaremos este procedimento ao seguinte trin\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-c85718d3d2ce230bbfb0ea503d218ad7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x^2+4x+3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -2px;\"><\/p>\n<\/p>\n<p> Calculamos os par\u00e2metros<\/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-14b463d0ecd5b350ced6cf1d6a12eef3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cf6d2c84f82625cb8a795ee1394251f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" 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-2fb8b0aec746411e81d4de8430957904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"h=\\cfrac{b}{2a}=\\cfrac{4}{2\\cdot 2} =1\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"142\" style=\"vertical-align: -12px;\"><\/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-84e97942072ca2139743f4cb2f853c44_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k=c-ah^2 = 3-2\\cdot 1^2 = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"214\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E, portanto, o polin\u00f4mio permanece: <\/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-00dc44f1798a5fab144056da5829a276_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2(x+1)^2+1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"101\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o da resolu\u00e7\u00e3o de todos os tipos de identidades not\u00e1veis (ou produtos not\u00e1veis). Voc\u00ea poder\u00e1 ver quais s\u00e3o as f\u00f3rmulas de todas as identidades not\u00e1veis, bem como exemplos e exerc\u00edcios resolvidos passo a passo. Al\u00e9m disso, mostraremos para que servem essas famosas regras matem\u00e1ticas. \ud83d\udc49\ud83d\udc49 Abaixo explicamos cada identidade not\u00e1vel passo &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Identidades not\u00e1veis (ou produtos not\u00e1veis)<\/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":[14],"tags":[],"class_list":["post-351","post","type-post","status-publish","format-standard","hentry","category-explicacoes-matematicas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Identidades not\u00e1veis (ou produtos not\u00e1veis) -<\/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\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Identidades not\u00e1veis (ou produtos not\u00e1veis) -\" \/>\n<meta property=\"og:description\" content=\"Aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o da resolu\u00e7\u00e3o de todos os tipos de identidades not\u00e1veis (ou produtos not\u00e1veis). Voc\u00ea poder\u00e1 ver quais s\u00e3o as f\u00f3rmulas de todas as identidades not\u00e1veis, bem como exemplos e exerc\u00edcios resolvidos passo a passo. Al\u00e9m disso, mostraremos para que servem essas famosas regras matem\u00e1ticas. \ud83d\udc49\ud83d\udc49 Abaixo explicamos cada identidade not\u00e1vel passo &hellip; Identidades not\u00e1veis (ou produtos not\u00e1veis) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T00:53:57+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/identites-produits-ou-egalites-notables.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=\"16 minutos\" 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