{"id":325,"date":"2023-07-06T07:57:24","date_gmt":"2023-07-06T07:57:24","guid":{"rendered":"https:\/\/mathority.org\/pt\/formula-para-o-produto-da-soma-pela-diferenca\/"},"modified":"2023-07-06T07:57:24","modified_gmt":"2023-07-06T07:57:24","slug":"formula-para-o-produto-da-soma-pela-diferenca","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/formula-para-o-produto-da-soma-pela-diferenca\/","title":{"rendered":"Produto da soma pela diferen\u00e7a (identidade not\u00e1vel)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula do produto da soma pela diferen\u00e7a. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de aplica\u00e7\u00e3o da f\u00f3rmula desse not\u00e1vel tipo de identidade, e ainda poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-el-producto-de-la-suma-por-la-diferencia\"><\/span> Qual \u00e9 o produto da soma e da diferen\u00e7a?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Em matem\u00e1tica, a no\u00e7\u00e3o de <strong>produto da soma pela diferen\u00e7a<\/strong> refere-se a uma das <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/identidades-produtos-igualdades-notaveis-exercicios-resolvidos\/\">igualdades not\u00e1veis<\/a><\/span><\/strong> , tamb\u00e9m chamadas de identidades not\u00e1veis ou produtos not\u00e1veis.<\/p>\n<p> Mais precisamente, a express\u00e3o para o produto da soma pela diferen\u00e7a tem a forma <strong>(a+b)\u00b7(ab)<\/strong> , onde (a+b) corresponde \u00e0 soma de dois termos diferentes, e (ab) \u00e9 a diferen\u00e7a desses mesmos dois termos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Formula-del-producto-de-la-suma-por-la-diferencia\"><\/span> F\u00f3rmula para produto de soma por diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agora que conhecemos a defini\u00e7\u00e3o matem\u00e1tica do produto da soma vezes a diferen\u00e7a, vamos ver qual f\u00f3rmula \u00e9 usada para resolver este not\u00e1vel tipo de identidade: <\/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=\"produto da soma pela diferen\u00e7a\" class=\"wp-image-2278\" width=\"255\" height=\"256\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, <strong>o produto da soma pela diferen\u00e7a de dois termos \u00e9 igual \u00e0 diferen\u00e7a dos quadrados desses termos<\/strong> . Em outras palavras, multiplicar a soma de dois termos diferentes pela subtra\u00e7\u00e3o desses mesmos dois termos equivale a elevar ao quadrado cada um dos 2 termos e subtra\u00ed-los.<\/p>\n<p> Isto implica que <a href=\"https:\/\/mathority.org\/pt\/diferenca-ou-subtracao-de-quadrados\/\"><strong><span style=\"text-decoration: underline;\">as diferen\u00e7as de quadrados<\/span><\/strong><\/a> podem ser fatoradas em produtos de somas vezes as diferen\u00e7as. Embora agora possa parecer complicado para voc\u00ea, na p\u00e1gina vinculada explicamos um truque que permite fatorar esse tipo de polin\u00f4mio em duas etapas simples. Clique e descubra como \u00e9 feito. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplos-de-productos-de-sumas-por-diferencias\"><\/span> Exemplos de produtos de somas por diferen\u00e7as<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de sabermos qual \u00e9 a f\u00f3rmula do produto da soma e da diferen\u00e7a, veremos a seguir v\u00e1rios exemplos resolvidos para que voc\u00ea possa entender melhor como se resolve esse not\u00e1vel tipo de igualdade.<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 1<\/h3>\n<ul>\n<li> Calcule, aplicando a f\u00f3rmula, o seguinte produto 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> A f\u00f3rmula para o produto da soma pela 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> Ent\u00e3o a primeira coisa que precisamos fazer \u00e9 identificar os valores 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> 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 agora que 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> Podemos aplicar 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<p>Como voc\u00ea pode ver, o produto de uma soma por uma diferen\u00e7a sempre dar\u00e1 um termo negativo. No entanto, isto n\u00e3o deve ser confundido com a not\u00e1vel identidade do quadrado de uma subtra\u00e7\u00e3o. Se voc\u00ea tiver alguma d\u00favida, recomendamos que d\u00ea uma olhada em qual \u00e9 a <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/quadrado-de-uma-diferenca-ou-subtracao\/\">f\u00f3rmula do quadrado da diferen\u00e7a<\/a><\/span><\/strong> , onde voc\u00ea tamb\u00e9m descobrir\u00e1 quais s\u00e3o as diferen\u00e7as entre essas duas identidades not\u00e1veis<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 2<\/h3>\n<ul>\n<li> Encontre, usando a f\u00f3rmula, o seguinte produto da soma pela diferen\u00e7a de dois bin\u00f4mios:<\/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-186f562faed63e636884263f5854f0e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3x+5)\\cdot (3x-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> A f\u00f3rmula para o produto da soma pela 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> Portanto, neste caso<\/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-723d3ff64e92fa43145366202ce30f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=3x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-b41cd8020624a683feb6c7e88b71c9ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\"><\/p>\n<p> . Portanto, se aplicarmos a f\u00f3rmula da soma por diferen\u00e7a, obteremos a seguinte express\u00e3o alg\u00e9brica:<\/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-fffde1dd457ab276c3d53e1d1e23616d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(3x+5)\\cdot (3x-5) = (3x)^2-5^2 = 9x^2-25\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"330\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"> Exemplo 3<\/h3>\n<ul>\n<li> Resolva com a f\u00f3rmula o seguinte produto da soma pela diferen\u00e7a de dois mon\u00f4mios:<\/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-409155e85d07804a9333bd48429940ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(4x-2y)\\cdot (4x+2y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"156\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como a multiplica\u00e7\u00e3o tem propriedade comutativa, multiplicar primeiro a diferen\u00e7a e depois a soma de duas quantidades equivale a multiplicar os mesmos par\u00eanteses ao contr\u00e1rio.<\/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-583e6cc781617efdb7061c82ae11ec0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(4x-2y)\\cdot (4x+2y) = (4x+2y)\\cdot (4x-2y)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"338\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, embora neste caso o produto seja invertido, ou seja, antes da adi\u00e7\u00e3o \u00e9 a subtra\u00e7\u00e3o, o resultado permanece o mesmo 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-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-72246fd1126cb58a3fb86376fba4f0d6_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> Ent\u00e3o neste problema<\/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-5231d0029f85ab0321bb3d4b9e25ec80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=4x\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" 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-f3fbd7a3d02814e8c9770e23ece10400_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b=2y\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -4px;\"><\/p>\n<p> . E quando tivermos identificado o valor de cada inc\u00f3gnita podemos usar a f\u00f3rmula para calcular o 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-3ff47675eb3b0a4db49ff32ebe471744_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(4x-2y)\\cdot (4x+2y)= (4x)^2-(2y)^2 = 16x^2-4y^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"389\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Demostracion-de-la-formula-de-la-suma-por-la-diferencia\"><\/span> Demonstra\u00e7\u00e3o da f\u00f3rmula de soma por diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A f\u00f3rmula da soma vezes a diferen\u00e7a que acabamos de estudar pode ser facilmente demonstrada.<\/p>\n<p> Se partirmos do produto de uma soma pela subtra\u00e7\u00e3o de quaisquer dois termos:<\/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> Simplesmente multiplique 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> E agrupando termos semelhantes, chegamos \u00e0 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-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> Portanto, a f\u00f3rmula para o produto not\u00e1vel da soma por diferen\u00e7as \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-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<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-del-producto-de-la-suma-por-diferencia\"><\/span> Exerc\u00edcios resolvidos para o produto da soma por diferen\u00e7a<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Abaixo preparamos v\u00e1rios exerc\u00edcios de adi\u00e7\u00e3o por diferen\u00e7as resolvidos passo a passo para que voc\u00ea possa praticar. Os exerc\u00edcios s\u00e3o ordenados do menos para o mais dif\u00edcil, por isso recomendamos come\u00e7ar com 1, continuar com 2 e finalmente fazer 3, que \u00e9 o mais dif\u00edcil.<\/p>\n<p> \u2b07\u2b07N\u00e3o se esque\u00e7a tamb\u00e9m que voc\u00ea pode nos deixar qualquer d\u00favida que surgir nos coment\u00e1rios!\u2b07\u2b07<\/p>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Resolva os seguintes 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-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b9857bf825403b9bb0b16fe08f338ac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l}(x+5)(x-5) = \\\\[2ex] =x^2-5^2=\\\\[2ex] = \\bm{x^2-25}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"167\" 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-7b16ddf395da600fb7bc3b5d62a328d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l}(2x+6)(2x-6) = \\\\[2ex] =(2x)^2-6^2=\\\\[2ex] = \\bm{4x^2-36}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"184\" 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-7109d4edfb32c8d194a747f30f23de59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l}(x+7)(x-7) = \\\\[2ex] =x^2-7^2=\\\\[2ex] = \\bm{x^2-49}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"166\" 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-ddc7c482ee40aa839c39f218036b2ac3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{array}{l}(x-4y)(x+4y) = \\\\[2ex] =(x+4y)(x-4y) =\\\\[2ex] =x^2-(4y)^2=\\\\[2ex] = \\bm{x^2-16y^2}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"130\" width=\"204\" 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> Expresse as seguintes multiplica\u00e7\u00f5es como diferen\u00e7as 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-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-ad159466aec4f0d240bd8d4c308f7abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\left(4x-5\\right)\\left(4x+5\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" 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-ce8c4004719f1aa6c92b21494efd621f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\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-6aa36e48b667ea4743daf1da540430ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\left(2x^3y+x^4y^2\\right)\\left(2x^3y-x^4y^2\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"248\" 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-130cd5d346f482730ecc3333446e523f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l}\\left(x^2+10\\right)\\left(x^2-10\\right) = \\\\[2ex] =\\left(x^2\\right)^2-10^2=\\\\[2ex] = \\bm{x^4-100}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"93\" width=\"208\" 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-789437ac865d440bbd7fdc8c3dcef649_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l}\\left(4x-5\\right)\\left(4x+5\\right) = \\\\[2ex] =(4x)^2-5^2=\\\\[2ex] = \\bm{16x^2-25}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"89\" width=\"187\" 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-95f3d6fb4f7ec97adeefac0c98b129a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l}\\left(8x^3+y^2\\right)\\left(8x^3-y^2\\right) = \\\\[2ex] =\\left(8x^3\\right)^2-\\left(y^2\\right)^2=\\\\[2ex] = \\bm{64x^6-y^4}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"224\" 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-cd84ca693626bc77a4a19e0c45a00caf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{D)} \\ \\begin{array}{l}\\left(2x^3y+x^4y^2\\right)\\left(2x^3y-x^4y^2\\right) = \\\\[2ex] =\\left(2x^3y\\right)^2-\\left(x^4y^2\\right)^2 =\\\\[2ex] = \\bm{4x^6y^2-x^8y^4}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"97\" width=\"279\" 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> Resolva 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-92f6afda6cffbe29df833c7579880aef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\left(9x^3+\\sqrt{5x}\\right)\\left(9x^3-\\sqrt{5x}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"227\" 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-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-2898ec593dbaadf26cda7de0359b147e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\left(6x^4+x^2+1\\right)\\left(6x^4-x^2-1\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"255\" 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 a primeira igualdade not\u00e1vel, \u00e9 preciso lembrar que uma raiz quadrada simplifica:<\/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-ab1362f27fa00986005f1065ad627537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{A)} \\ \\begin{array}{l}\\left(9x^3+\\sqrt{5x}\\right)\\left(9x^3-\\sqrt{5x}\\right) = \\\\[2ex] =\\left(9x^3\\right)^2-\\left(\\sqrt{5x}\\right)^2=\\\\[2ex] = \\bm{81x^6-5x}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"94\" width=\"258\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os 2 mon\u00f4mios da segunda soma por diferen\u00e7a possuem coeficientes fracion\u00e1rios, portanto devemos resolver este exerc\u00edcio utilizando 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-f326d1d0330f288f08c8cf6a27ba85e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{B)} \\ \\begin{array}{l}\\displaystyle \\left(\\frac{1}{2}x^2+\\frac{5}{3}x\\right)\\left(\\frac{1}{2}x^2-\\frac{5}{3}x\\right) = \\\\[4ex] \\displaystyle =\\left(\\frac{1}{2}x^2\\right)^2-\\left(\\frac{5}{3}x\\right)^2=\\\\[4ex] \\displaystyle =\\frac{1^2}{2^2}x^4-\\frac{5^2}{3^2}x^2=\\\\[4ex]\\displaystyle = \\mathbf{\\frac{1}{4}}\\bm{x^4-}\\mathbf{\\frac{25}{9}}\\bm{x^2} \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"228\" width=\"263\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, a \u00faltima igualdade not\u00e1vel \u00e9 um pouco especial porque cont\u00e9m outro produto not\u00e1vel (o quadrado da 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-ba1748a8a88df231365c5134f251038a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{C)} \\ \\begin{array}{l}\\left(6x^4+x^2+1\\right)\\left(6x^4-x^2-1\\right) = \\\\[2ex] = \\left(6x^4+\\left(x^2+1\\right)\\right)\\left(6x^4-\\left(x^2+1\\right)\\right)=\\\\[2ex]=\\left(6x^4\\right)^2-\\left(x^2+1\\right)^2=\\\\[2ex] =36x^8 - \\left(x^4+2x^2+1\\right)=\\\\[2ex] = \\bm{36x^8 - x^4-2x^2-1}\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"167\" width=\"338\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula do produto da soma pela diferen\u00e7a. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de aplica\u00e7\u00e3o da f\u00f3rmula desse not\u00e1vel tipo de identidade, e ainda poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 o produto da soma e da diferen\u00e7a? Em matem\u00e1tica, a no\u00e7\u00e3o de produto da soma pela &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/formula-para-o-produto-da-soma-pela-diferenca\/\"> <span class=\"screen-reader-text\">Produto da soma pela diferen\u00e7a (identidade not\u00e1vel)<\/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":[15],"tags":[],"class_list":["post-325","post","type-post","status-publish","format-standard","hentry","category-identidades-notaveis"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Produto da soma pela diferen\u00e7a (identidade not\u00e1vel) - Matoridade<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/pt\/formula-para-o-produto-da-soma-pela-diferenca\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Produto da soma pela diferen\u00e7a (identidade not\u00e1vel) - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a f\u00f3rmula do produto da soma pela diferen\u00e7a. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de aplica\u00e7\u00e3o da f\u00f3rmula desse not\u00e1vel tipo de identidade, e ainda poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 o produto da soma e da diferen\u00e7a? 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