{"id":290,"date":"2023-07-06T18:20:29","date_gmt":"2023-07-06T18:20:29","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/"},"modified":"2023-07-06T18:20:29","modified_gmt":"2023-07-06T18:20:29","slug":"exemplos-de-polinomios-homogeneos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/","title":{"rendered":"Polin\u00f4mio homog\u00eaneo"},"content":{"rendered":"<p>Esta p\u00e1gina explica o que s\u00e3o polin\u00f4mios homog\u00eaneos. Voc\u00ea tamb\u00e9m ver\u00e1 exemplos de polin\u00f4mios homog\u00eaneos e as propriedades desse tipo de polin\u00f4mio. E, al\u00e9m disso, voc\u00ea encontrar\u00e1 a diferen\u00e7a entre polin\u00f4mios homog\u00eaneos e polin\u00f4mios heterog\u00eaneos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-un-polinomio-homogeneo\"><\/span> O que \u00e9 um polin\u00f4mio homog\u00eaneo?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A defini\u00e7\u00e3o de um polin\u00f4mio homog\u00eaneo \u00e9 a seguinte:<\/p>\n<p> <strong>Em matem\u00e1tica, um polin\u00f4mio homog\u00eaneo \u00e9 um polin\u00f4mio em que todos os termos t\u00eam o mesmo grau.<\/strong><\/p>\n<p> Um exemplo de polin\u00f4mio homog\u00eaneo seria:<\/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-9c562859bfcb9245bf750cdf62e30cc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x,y,z)=x^3+5x^2y-4xyz\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"230\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Neste caso, trata-se de um polin\u00f4mio homog\u00eaneo de grau 3, pois todos os mon\u00f4mios que fazem parte do polin\u00f4mio s\u00e3o de terceiro grau.<\/p>\n<p> Se voc\u00ea tiver alguma d\u00favida sobre como \u00e9 calculado o grau de um termo de um polin\u00f4mio homog\u00eaneo, pode consultar nossa p\u00e1gina sobre <a href=\"https:\/\/mathority.org\/pt\/partes-de-um-coeficiente-monomio-parte-literal-e-grau-variavel\/\"><strong><span style=\"text-decoration: underline;\">quais s\u00e3o as partes de um mon\u00f4mio<\/span><\/strong><\/a> , onde voc\u00ea n\u00e3o s\u00f3 encontrar\u00e1 como encontrar o grau de um mon\u00f4mio, mas tamb\u00e9m a explica\u00e7\u00e3o de todas as partes de um mon\u00f4mio e como identific\u00e1-las. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplos-de-polinomios-homogeneos\"><\/span> Exemplos de polin\u00f4mios homog\u00eaneos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Depois de vermos o que significa um polin\u00f4mio ser homog\u00eaneo, vejamos alguns exemplos de polin\u00f4mios homog\u00eaneos para finalizar a compreens\u00e3o do conceito:<\/p>\n<ul>\n<li> Exemplo de um polin\u00f4mio homog\u00eaneo de grau 5:<\/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-938803398f7974d4ff410dfb1e43f84e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x,y)=x^5+3x^2y^3-6x^4y+10xy^4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Exemplo de um polin\u00f4mio homog\u00eaneo de grau 7:<\/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-b24c1a5bf1ed25c5698108cc448bdb4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x,y,z)=x^3y^4+2x^5y^2+4x^2y^2z^3-x^2y^4z\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"343\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Exemplo de um polin\u00f4mio homog\u00eaneo de grau 13: <\/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-50fee4914e8d236a46dbc69f8d67b3d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(a,b,c)=7a^6b^4c^3+2a^8b^3c^2+5a^4b^8c\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"297\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Polinomio-homogeneo-y-polinomio-heterogeneo\"><\/span> Polin\u00f4mio homog\u00eaneo e polin\u00f4mio heterog\u00eaneo<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Deve-se notar que outro polin\u00f4mio muito semelhante ao polin\u00f4mio homog\u00eaneo \u00e9 o polin\u00f4mio heterog\u00eaneo, embora exista uma diferen\u00e7a fundamental entre eles:<\/p>\n<p> Um <strong>polin\u00f4mio heterog\u00eaneo<\/strong> \u00e9 um polin\u00f4mio em que todos os termos n\u00e3o t\u00eam o mesmo grau.<\/p>\n<p> Portanto, somente quando um mon\u00f4mio do polin\u00f4mio tiver grau diferente do resto dos elementos, esse polin\u00f4mio ser\u00e1 heterog\u00eaneo.<\/p>\n<p> Por exemplo, o seguinte polin\u00f4mio \u00e9 heterog\u00eaneo:<\/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-8baa087cc08645b03a7879dc38522298_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x,y)=x^4+2x^3y+8x^2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"202\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Embora dois dos termos do polin\u00f4mio sejam de grau 4 (x <sup>4<\/sup> , 2x <sup>3<\/sup> y), na verdade \u00e9 um polin\u00f4mio heterog\u00eaneo porque possui outro termo de grau diferente (8x <sup>2<\/sup> \u00e9 de grau 2).<\/p>\n<p> Como voc\u00ea pode ver, polin\u00f4mios homog\u00eaneos e heterog\u00eaneos s\u00e3o muito semelhantes entre si e facilmente confundidos, por isso precisamos ter cuidado. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Propiedades-de-los-polinomios-homogeneos\"><\/span> Propriedades de polin\u00f4mios homog\u00eaneos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Polin\u00f4mios homog\u00eaneos t\u00eam as seguintes caracter\u00edsticas<\/p>\n<ul>\n<li> O n\u00famero de diferentes mon\u00f4mios homog\u00eaneos de grau M em um polin\u00f4mio de N vari\u00e1veis pode ser calculado usando a seguinte f\u00f3rmula:<\/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-a50a9af76c5b398ccf1a017a08eb728c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{(M+N-1)!}{M!(N-1)!}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"107\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Talvez o \u201c <strong>!<\/strong> \u201d \u00bb parece estranho para voc\u00ea que seja usado em \u00e1lgebra. Bem, voc\u00ea deve saber que \u00e9 usado para indicar uma opera\u00e7\u00e3o matem\u00e1tica especial, chamada <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/funcao-fatorial-de-um-numero\/\">fatorial de um n\u00famero<\/a><\/span><\/strong> . Voc\u00ea pode ver em que consiste esta opera\u00e7\u00e3o e para que serve no link anterior.<\/p>\n<ul>\n<li> A express\u00e3o para a s\u00e9rie de Taylor que corresponde a um polin\u00f4mio homog\u00eaneo estendido no ponto <em>x<\/em> \u00e9 a seguinte:<\/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-8f4fe67ef967d0006f8ec3f8936d3f1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x+y)= \\sum_{j=0}^n {n \\choose j}  \\check{P} (\\underbrace{x,x,\\dots ,x}_{j} &amp; \\underbrace{y,y,\\dots ,y}_{n-j})\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"346\" style=\"vertical-align: -30px;\"><\/p>\n<\/p>\n<p> Por\u00e9m, para poder aplicar (e compreender) esta propriedade, \u00e9 necess\u00e1rio saber como a express\u00e3o \u00e9 calculada<\/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-4c062127739b922dff24815ba6bb3660_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{pmatrix} n \\\\ j \\end{pmatrix} ,\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"41\" style=\"vertical-align: -17px;\"><\/p>\n<p> chamado de n\u00famero combinat\u00f3rio. Portanto, se voc\u00ea n\u00e3o entendeu a propriedade anterior, recomendo que veja qual \u00e9 a <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/propriedades-do-coeficiente-binomial-de-numero-combinatorio\/\">f\u00f3rmula do n\u00famero combinat\u00f3rio<\/a><\/span><\/strong> .<\/p>\n<div id=\"ezoic-pub-ad-placeholder-176\" data-inserter-version=\"-1\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Esta p\u00e1gina explica o que s\u00e3o polin\u00f4mios homog\u00eaneos. Voc\u00ea tamb\u00e9m ver\u00e1 exemplos de polin\u00f4mios homog\u00eaneos e as propriedades desse tipo de polin\u00f4mio. E, al\u00e9m disso, voc\u00ea encontrar\u00e1 a diferen\u00e7a entre polin\u00f4mios homog\u00eaneos e polin\u00f4mios heterog\u00eaneos. O que \u00e9 um polin\u00f4mio homog\u00eaneo? A defini\u00e7\u00e3o de um polin\u00f4mio homog\u00eaneo \u00e9 a seguinte: Em matem\u00e1tica, um polin\u00f4mio homog\u00eaneo &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\"> <span class=\"screen-reader-text\">Polin\u00f4mio homog\u00eaneo<\/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":[38],"tags":[],"class_list":["post-290","post","type-post","status-publish","format-standard","hentry","category-tipos-de-polinomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Polin\u00f4mio homog\u00eaneo - 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\/exemplos-de-polinomios-homogeneos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Polin\u00f4mio homog\u00eaneo - Matoridade\" \/>\n<meta property=\"og:description\" content=\"Esta p\u00e1gina explica o que s\u00e3o polin\u00f4mios homog\u00eaneos. Voc\u00ea tamb\u00e9m ver\u00e1 exemplos de polin\u00f4mios homog\u00eaneos e as propriedades desse tipo de polin\u00f4mio. E, al\u00e9m disso, voc\u00ea encontrar\u00e1 a diferen\u00e7a entre polin\u00f4mios homog\u00eaneos e polin\u00f4mios heterog\u00eaneos. O que \u00e9 um polin\u00f4mio homog\u00eaneo? A defini\u00e7\u00e3o de um polin\u00f4mio homog\u00eaneo \u00e9 a seguinte: Em matem\u00e1tica, um polin\u00f4mio homog\u00eaneo &hellip; Polin\u00f4mio homog\u00eaneo Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T18:20:29+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9c562859bfcb9245bf750cdf62e30cc4_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Polin\u00f4mio homog\u00eaneo\",\"datePublished\":\"2023-07-06T18:20:29+00:00\",\"dateModified\":\"2023-07-06T18:20:29+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\"},\"wordCount\":568,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Tipos de polin\u00f4mios\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\",\"url\":\"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/\",\"name\":\"Polin\u00f4mio homog\u00eaneo - 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Matoridade","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/exemplos-de-polinomios-homogeneos\/","og_locale":"pt_BR","og_type":"article","og_title":"Polin\u00f4mio homog\u00eaneo - Matoridade","og_description":"Esta p\u00e1gina explica o que s\u00e3o polin\u00f4mios homog\u00eaneos. Voc\u00ea tamb\u00e9m ver\u00e1 exemplos de polin\u00f4mios homog\u00eaneos e as propriedades desse tipo de polin\u00f4mio. E, al\u00e9m disso, voc\u00ea encontrar\u00e1 a diferen\u00e7a entre polin\u00f4mios homog\u00eaneos e polin\u00f4mios heterog\u00eaneos. O que \u00e9 um polin\u00f4mio homog\u00eaneo? 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