{"id":55,"date":"2023-09-17T07:25:59","date_gmt":"2023-09-17T07:25:59","guid":{"rendered":"https:\/\/mathority.org\/pt\/raizes-de-um-polinomio\/"},"modified":"2023-09-17T07:25:59","modified_gmt":"2023-09-17T07:25:59","slug":"raizes-de-um-polinomio","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/raizes-de-um-polinomio\/","title":{"rendered":"Ra\u00edzes de um polin\u00f4mio"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o as ra\u00edzes de um polin\u00f4mio e como elas s\u00e3o calculadas. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo nas ra\u00edzes de um polin\u00f4mio. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-son-las-raices-de-un-polinomio\"><\/span> Quais s\u00e3o as ra\u00edzes de um polin\u00f4mio?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> <strong>Em matem\u00e1tica, as ra\u00edzes (ou zeros) de um polin\u00f4mio s\u00e3o os valores que cancelam o polin\u00f4mio. Ou seja, as ra\u00edzes de um polin\u00f4mio s\u00e3o todos aqueles valores que, quando avaliados no polin\u00f4mio, possuem valor num\u00e9rico igual a 0.<\/strong><\/p>\n<p> Eventualmente,<\/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-2b24e8b3f28f048c85d6ea0f32d59fff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma raiz do polin\u00f4mio<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a80be6e42ac3b3c6528958bbfa21f92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<p> Sim <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-05f8ce1400e1aeecb7fc4e9548e2c5e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(a)=0.\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"74\" style=\"vertical-align: -5px;\"><\/p>\n<\/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\/racines-ou-zeros-dun-polynome.png\" alt=\"ra\u00edzes ou zeros de um polin\u00f4mio\" class=\"wp-image-916\" width=\"170\" height=\"172\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<\/div>\n<p> Por exemplo, se tivermos o seguinte polin\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-9f4ac06469282d1968cf43d2d7dc35ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2-3x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"150\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Podemos verificar que uma das ra\u00edzes do polin\u00f4mio \u00e9 1, pois o valor num\u00e9rico do polin\u00f4mio em <em>x=1<\/em> \u00e9 igual a zero:<\/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-30a608c13e6fb189405ac92258df7e3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1)=1^2-3\\cdot 1+2 = 1-3+2 \\color{blue} \\bm{= 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"312\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Por outro lado, 3 n\u00e3o \u00e9 raiz do polin\u00f4mio porque n\u00e3o \u00e9 um valor que anula o polin\u00f4mio, ou seja, o valor num\u00e9rico do polin\u00f4mio em <em>x=3<\/em> \u00e9 diferente de zero:<\/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-b093c5f3d6e2bac6c79c9c0a73182f39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3)=3^2-3\\cdot 3+2 = 9-9+2=2  \\color{blue} \\bm{\\neq  0}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"345\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Voc\u00ea provavelmente agora entende melhor o que \u00e9 a raiz de um polin\u00f4mio, mas n\u00e3o gostaria de saber quantas ra\u00edzes um polin\u00f4mio tem? Ou como encontrar todas as ra\u00edzes de um polin\u00f4mio? Bem, \u00e9 exatamente isso que veremos na pr\u00f3xima se\u00e7\u00e3o. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFComo-calcular-todas-las-raices-de-un-polinomio\"><\/span> Como calcular todas as ra\u00edzes de um polin\u00f4mio?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para encontrar todas as ra\u00edzes de um polin\u00f4mio, voc\u00ea deve seguir os seguintes passos:<\/p>\n<ol style=\"color:#ff5733; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Primeiro, todos os divisores do termo independente do polin\u00f4mio s\u00e3o calculados.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Segundo, todos os valores encontrados na etapa anterior s\u00e3o avaliados no polin\u00f4mio.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">Por fim, se ao avaliar um n\u00famero no polin\u00f4mio seu valor num\u00e9rico for igual a zero, esse n\u00famero \u00e9 raiz do polin\u00f4mio. Caso contr\u00e1rio, o referido n\u00famero n\u00e3o corresponde \u00e0 raiz do polin\u00f4mio.<\/span><\/li>\n<\/ol>\n<p> Este procedimento \u00e9 deduzido do <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/exemplos-e-exercicios-do-teorema-do-resto-resolvidos\/\">teorema do resto<\/a><\/span><\/strong> , clique neste link para descobrir o motivo deste procedimento espec\u00edfico. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejemplo-de-como-se-calculan-las-raices-de-un-polinomio\"><\/span> Exemplo de c\u00e1lculo das ra\u00edzes de um polin\u00f4mio<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A seguir resolveremos um exemplo passo a passo para que voc\u00ea entenda melhor como tirar as ra\u00edzes de um polin\u00f4mio.<\/p>\n<ul>\n<li> Quais s\u00e3o todas as ra\u00edzes do seguinte polin\u00f4mio?<\/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-18b4f499034ee1404f872bd26694996e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = x^2-5x+6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"151\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em primeiro lugar, devemos encontrar os divisores do termo independente, pois toda raiz de um polin\u00f4mio \u00e9 tamb\u00e9m um divisor do termo independente. Ent\u00e3o, os divisores de 6 s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> Divisores de 6: +1, -1, +2, -2, +3, -3<\/p>\n<p> Lembre-se que se um n\u00famero \u00e9 um divisor, seu negativo tamb\u00e9m \u00e9 um divisor. Como um n\u00famero \u00e9 divis\u00edvel por n\u00fameros positivos e negativos.<\/p>\n<p> Assim, as poss\u00edveis ra\u00edzes ou zeros do polin\u00f4mio s\u00e3o: \u00b11, \u00b12, \u00b13. Portanto, precisamos determinar o valor num\u00e9rico do polin\u00f4mio para todos esses valores. E, para isso, substitu\u00edmos esses valores na express\u00e3o do polin\u00f4mio onde existe um x: <\/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-16fa59fcdbbaa92788e0292f40f15365_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1) = 1^2 -5\\cdot 1 +6= 1 -5 +6 =2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"285\" 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-3e0acf6e160800e125f3efb9799e1cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1) = (-1)^2 -5\\cdot (-1) +6 =1+5+6 = 12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"363\" 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-aad7d84847b59d9fa4d3ad780e20a5d4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2) = 2^2 -5\\cdot 2 +6 =4-10+6= \\color{blue} \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"326\" 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-da1cb9edee6182d6abc4df0155acb975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2) = (-2)^2 -5\\cdot (-2) +6 =4+10+6 =20\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"372\" 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-4122327630c5bd1d2835ff1ab612ab00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3) = 3^2 -5\\cdot 3 +6 =9-15+6=\\color{blue} \\bm{0}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"326\" 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-6fc3fe1dc37a53aa473b0ca773a5f500_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-3) = (-3)^2 -5\\cdot (-3) +6 =9+15+6 =30\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"372\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o o polin\u00f4mio s\u00f3 desaparece quando a vari\u00e1vel <em>x<\/em> \u00e9 +2 ou +3, ent\u00e3o aqui est\u00e3o as ra\u00edzes do polin\u00f4mio:<\/p>\n<p class=\"has-text-align-center\"> <strong>Ra\u00edzes ou zeros do polin\u00f4mio<\/strong> : +2 e +3<\/p>\n<p> Por outro lado, observe que o polin\u00f4mio tem tantas ra\u00edzes quanto o seu grau, ou seja, como o polin\u00f4mio \u00e9 de segundo grau, ele tem duas ra\u00edzes. Nas propriedades das ra\u00edzes de um polin\u00f4mio (abaixo), veremos porque essa caracter\u00edstica sempre vale para qualquer polin\u00f4mio.<\/p>\n<p> Acabamos de ver uma maneira de determinar as ra\u00edzes de um polin\u00f4mio. No entanto, ainda existem outros m\u00e9todos para conseguir isso, por exemplo, voc\u00ea tamb\u00e9m pode encontrar as ra\u00edzes de um polin\u00f4mio com a regra de Ruffini. Clique no link a seguir para ver <strong><span style=\"text-decoration: underline;\"><a href=\"https:\/\/mathority.org\/pt\/regras-resolvidas-exemplos-exercicios-ruffini\/\">exemplos da regra de Ruffini<\/a><\/span><\/strong> , aqui voc\u00ea descobrir\u00e1 em que consiste esse conhecido m\u00e9todo e, tamb\u00e9m, quais as diferen\u00e7as entre os dois procedimentos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Propiedades-de-las-raices-de-un-polinomio\"><\/span> Propriedades das ra\u00edzes de um polin\u00f4mio<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> As ra\u00edzes ou zeros de um polin\u00f4mio possuem as seguintes caracter\u00edsticas:<\/p>\n<ol start=\"1\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Como vimos anteriormente, as ra\u00edzes inteiras (ou zeros) de um polin\u00f4mio s\u00e3o divisores do termo independente do polin\u00f4mio.<\/span><\/li>\n<\/ol>\n<ol start=\"2\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Se conhecermos todas as ra\u00edzes de um polin\u00f4mio, podemos expressar esse polin\u00f4mio na forma de produtos de bin\u00f4mios do tipo\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ab8c02c3b91a39a9d1a155e9d0c5fa93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x-a).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<\/ol>\n<p> Por exemplo, o polin\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-aa4ef60979e504a668114218a4258c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3+3x^2-x-3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"190\" style=\"vertical-align: -5px;\"><\/p>\n<p> Possui 3 ra\u00edzes que s\u00e3o<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3a62eca4d3d0e41d5d4b43e484a9b451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1, x=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"120\" 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-18062540cd799901f80ebaea09891a13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-3.\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"61\" style=\"vertical-align: 0px;\"><\/p>\n<p> Podemos, portanto, reescrever o polin\u00f4mio na forma de 3 multiplica\u00e7\u00f5es de fatores, cada uma formada pela 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 uma raiz mudou de sinal:<\/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-ce071610927d2723c8ac2e7b299c1c5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\definecolor{vermell}{HTML}{F44336}\\definecolor{blau}{HTML}{2196F3}\\definecolor{verd}{HTML}{27AE60} P(x) =x^3+3x^2-x-3 \\ \\longrightarrow \\ \\text{ra\\'ices} \\begin{cases} x=\\color{verd}\\bm{+1} \\\\[2ex] x=\\color{vermell}\\bm{-1} \\\\[2ex] x=\\color{blau}\\bm{-3}\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"131\" width=\"609\" 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-a69abcf91f1dec9f01082f2d5866fa01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{vermell}{HTML}{F44336}\\definecolor{blau}{HTML}{2196F3}\\definecolor{verd}{HTML}{27AE60}P(x) =x^3+3x^2-x-3 = (x\\color{verd}\\bm{-1}\\color{black})\\cdot (x\\color{vermell}\\bm{+1}\\color{black}) \\cdot (x\\color{blau}\\bm{+3}\\color{black})\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"582\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Isso \u00e9 chamado de fatora\u00e7\u00e3o polinomial. Na verdade, uma das principais aplica\u00e7\u00f5es da determina\u00e7\u00e3o das ra\u00edzes de um polin\u00f4mio \u00e9 que elas s\u00e3o usadas para fator\u00e1-lo. No link a seguir voc\u00ea poder\u00e1 descobrir em que consiste essa opera\u00e7\u00e3o t\u00e3o especial e, al\u00e9m disso, poder\u00e1 praticar com <a href=\"https:\/\/mathority.org\/pt\"><strong><span style=\"text-decoration: underline;\">exerc\u00edcios resolvidos de fatora\u00e7\u00e3o polinomial<\/span><\/strong><\/a> .<\/p>\n<ol start=\"3\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Um polin\u00f4mio tem tantas ra\u00edzes quanto seu grau indica. Portanto, um polin\u00f4mio de segundo grau ter\u00e1 2 ra\u00edzes, um polin\u00f4mio de terceiro grau ter\u00e1 3 ra\u00edzes, um polin\u00f4mio de quarto grau ter\u00e1 4 ra\u00edzes e assim por diante.<\/span><\/li>\n<\/ol>\n<ol start=\"4\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Se um polin\u00f4mio n\u00e3o possui um termo independente, significa que uma de suas ra\u00edzes \u00e9 0. Ent\u00e3o as demais ra\u00edzes devem ser divisores do coeficiente do mon\u00f4mio de menor grau.<\/span><\/li>\n<\/ol>\n<p> Por exemplo, o seguinte polin\u00f4mio n\u00e3o possui termo independente:<\/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-06bfb6282e808e7365c497c01ff60eee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^3+x^2-2x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"159\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Portanto, uma raiz do polin\u00f4mio deve necessariamente ser 0. E o restante das ra\u00edzes s\u00e3o divisores do coeficiente do termo de grau mais baixo, ou seja, -2. Mais precisamente, as outras ra\u00edzes s\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-eec86cbca5afb38459e47b0dce5eb23a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=+1\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"56\" style=\"vertical-align: -2px;\"><\/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-ffba38287436639a3011d50b97654cd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-2,\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\"><\/p>\n<p> ent\u00e3o todas as ra\u00edzes do polin\u00f4mio s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> Ra\u00edzes ou zeros do polin\u00f4mio: 0, +1 e -2<\/p>\n<ol start=\"5\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Quando as ra\u00edzes de um polin\u00f4mio n\u00e3o podem ser determinadas, diz-se que \u00e9 um polin\u00f4mio irredut\u00edvel.<\/span><\/li>\n<\/ol>\n<p> Por exemplo, tentaremos calcular as ra\u00edzes do seguinte polin\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-4d20f05e1f8e2433a542f83bfaf519e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) =x^2+3x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"150\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> As \u00fanicas ra\u00edzes poss\u00edveis do polin\u00f4mio s\u00e3o os divisores de -1, ou seja -1 e +1. Portanto, avaliamos o polin\u00f4mio com estes valores:<\/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-c1128c014be9c5f642dd8a249c0fc7bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1) = 1^2 +3\\cdot 1 -1= 1 +3 -1 =3 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"318\" 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-15853e7e7134e2b54d4470568ea6b92a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1) = (-1)^2 +3\\cdot (-1)-1 =1-3-1 =-3 \\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"401\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Em nenhum caso o polin\u00f4mio \u00e9 cancelado, portanto n\u00e3o possui ra\u00edzes e, portanto, \u00e9 um polin\u00f4mio irredut\u00edvel.<\/p>\n<ol start=\"6\" style=\"color:#ff5733; font-weight: bold;\">\n<li> <span style=\"color:#0c0c0c;font-weight: normal;\">Quando o polin\u00f4mio \u00e9 composto pelo produto de v\u00e1rios polin\u00f4mios, n\u00e3o \u00e9 necess\u00e1rio fazer este produto para calcular as ra\u00edzes, mas as ra\u00edzes do polin\u00f4mio s\u00e3o as ra\u00edzes de cada fator multiplicadas.<\/span><\/li>\n<\/ol>\n<p> Por exemplo, se tivermos o seguinte polin\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-c5b7b6a31a96b6ed2e2e0187ea6aa8ba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x) = (x-2) \\cdot (x+1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"182\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Da segunda propriedade das ra\u00edzes dos polin\u00f4mios, podemos deduzir que a raiz do polin\u00f4mio esquerdo \u00e9 +2 e a raiz do polin\u00f4mio direito \u00e9 -1.<\/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-18a96a84b626c71b099da0c446b6367f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (x-2) \\ \\longrightarrow \\ \\text{ra\\'iz} \\ x=+2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-10dab76581eedafb14febe2a82f03005_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (x+1) \\ \\longrightarrow \\ \\text{ra\\'iz} \\ x=-1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Assim, as ra\u00edzes do polin\u00f4mio resultante da multiplica\u00e7\u00e3o dos dois fatores s\u00e3o as suas respectivas ra\u00edzes, ou seja, +2 e -1. <\/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-416399918b5a2a051a6bfc7343ef7960_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  P(x) = (x-2) \\cdot (x+1) \\ \\longrightarrow \\ \\text{ra\\'ices} \\ \\begin{cases}x=+2  \\\\[2ex] x=-1 \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"357\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicios-resueltos-de-raices-de-polinomios\"><\/span> Exerc\u00edcios resolvidos sobre ra\u00edzes de polin\u00f4mios<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine se<\/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-4c11e65374dfa6f887fb53ffc5765aed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x = -4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma raiz do seguinte polin\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-af9bc84f96cb9f496322a41fe8818906_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3+2x^2-11x-12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para descobrir se<\/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-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 uma raiz do polin\u00f4mio, precisamos avali\u00e1-lo com esse valor. Ainda:<\/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-0c88c4456693b0c57d55aba68287414c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}P(-4)&amp; =(-4)^3+2\\cdot (-4)^2-11\\cdot (-4) -12 \\\\[2ex] &amp; = -64+2\\cdot 16 +44 -12 \\\\[2ex] &amp; = -64+32+44 -12 \\\\[2ex] &amp; = 0 \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"142\" width=\"333\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O valor num\u00e9rico do polin\u00f4mio em<\/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-b3dc975a98ccada6f136856736d7df06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 zero, ent\u00e3o \u00e9 efetivamente uma raiz do polin\u00f4mio.<\/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> Calcule todas as ra\u00edzes do seguinte polin\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-9f4ac06469282d1968cf43d2d7dc35ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^2-3x+2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"150\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro, para encontrar as poss\u00edveis ra\u00edzes do polin\u00f4mio, devemos encontrar os divisores do termo independente. Ent\u00e3o, os divisores de 2 s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> Divisores de 2: +1, -1, +2, -2<\/p>\n<p class=\"has-text-align-left\"> As poss\u00edveis ra\u00edzes ou zeros do polin\u00f4mio s\u00e3o, portanto, \u00b11 e \u00b12. Portanto, precisamos calcular quanto o polin\u00f4mio est\u00e1 em todos esses valores: <\/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-41fba5390c3cecf1515f0a03890981a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1)=1^2-3\\cdot 1+2 =1-3+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"286\" 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-c8bfa6157619295c166a356db7b6fd1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1)=(-1)^2-3\\cdot (-1)+2 =1+3+2=6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"355\" 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-b6c2ccc218c0f8e8ffd9eb599a0063b0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2)=2^2-3\\cdot 2+2 =4-6+2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"286\" 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-d10352d63709d5958d81f4910c0bf9cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=(-2)^2-3\\cdot (-2)+2 =4+6+2=12\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"363\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, o polin\u00f4mio desaparece quando <em>x<\/em> \u00e9 +1 ou +2, ent\u00e3o aqui est\u00e3o as ra\u00edzes do polin\u00f4mio:<\/p>\n<p class=\"has-text-align-center\"> <strong>Ra\u00edzes ou zeros do polin\u00f4mio<\/strong> : +1 e +2<\/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> Encontre as ra\u00edzes do seguinte polin\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-dd0105a18a9affad36c35065a8460095_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3-x^2-4x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"190\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Devemos primeiro encontrar os divisores do termo independente, pois a raiz de um polin\u00f4mio tamb\u00e9m \u00e9 um divisor do termo independente. Ent\u00e3o, os divisores de 4 s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> Divisores de 4: +1, -1, +2, -2, +4, -4<\/p>\n<p class=\"has-text-align-left\"> As poss\u00edveis ra\u00edzes ou zeros do polin\u00f4mio s\u00e3o, portanto, \u00b11, \u00b12 e \u00b14. Devemos, portanto, encontrar o valor num\u00e9rico do polin\u00f4mio em todos estes valores: <\/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-49c2352542c6a029b23b2f23f505e2d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1)=1^3-1^2-4\\cdot 1+4  =1-1-4+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"354\" 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-1d1819dce07b117e3afc654b4c3b6f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1)=(-1)^3-(-1)^2-4\\cdot (-1)+4 =-1-1+4+4=6\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"465\" 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-1732090f2003d044169d6fec895e790e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2)=2^3-2^2-4\\cdot 2+4 =8-4-8+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"354\" 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-d825dc0a2c377ab13e3c1b404252d2eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=(-2)^3-(-2)^2-4\\cdot (-2)+4 =-8-4+8+4=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"465\" 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-eacdec999ade2fcddaa431651bb5a24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3)=3^3-3^2-4\\cdot 3+4 =27-9-12+4=10\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"381\" 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-d9db73e42d678a731d3343d3b9d61aad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-3)=(-3)^3-(-3)^2-4\\cdot (-3)+4 =-27-9+12+4=20\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"491\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, o polin\u00f4mio s\u00f3 desaparece quando <em>x<\/em> \u00e9 +1, +2 ou -2, ent\u00e3o aqui est\u00e3o as ra\u00edzes do polin\u00f4mio:<\/p>\n<p class=\"has-text-align-center\"> <strong>Ra\u00edzes ou zeros do polin\u00f4mio<\/strong> : +1, +2 e -2<\/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> Encontre as ra\u00edzes do seguinte polin\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-dc922d3e38a3393ff70f5f382ea9518a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=x^3-6x^2+8x\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"168\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Neste caso, o polin\u00f4mio n\u00e3o possui termo independente. Portanto, de acordo com a quarta propriedade das ra\u00edzes explicada acima, sabemos que uma das ra\u00edzes do polin\u00f4mio deve ser 0.<\/p>\n<p class=\"has-text-align-center\"> Ra\u00edzes do polin\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-8203ced39e0cdafefa708857c7ec2264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, neste caso, as ra\u00edzes poss\u00edveis n\u00e3o s\u00e3o os divisores do termo independente, mas sim as do coeficiente do termo de menor grau, ou seja, 8:<\/p>\n<p class=\"has-text-align-center\"> Divisores de 8: +1, -1, +2, -2, +4, -4, +8, -8<\/p>\n<p class=\"has-text-align-left\"> Portanto, as poss\u00edveis ra\u00edzes ou zeros do polin\u00f4mio s\u00e3o \u00b11, \u00b12, \u00b14 e \u00b18. Devemos, portanto, calcular o valor num\u00e9rico do polin\u00f4mio em todos estes valores: <\/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-5924a559a5d5fda73011bcc015baf065_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1)=1^3-6\\cdot 1^2+8\\cdot 1 = 1-6+8=3\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"315\" 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-f32e3cf981a0aadf450398e01009ca32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1)=(-1)^3-6\\cdot (-1)^2+8\\cdot (-1) = -1-6-8=-15\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"447\" 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-cf97abc29d59bc2bd3ccd23c0dc4c65e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(2)=2^3-6\\cdot 2^2+8\\cdot 2 = 8-24+16=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"333\" 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-bfbaf3ddbb35e0e6b0485f470b1d728e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2)=(-2)^3-6\\cdot (-2)^2+8\\cdot (-2) = -8-24-16=-48\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"466\" 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-e0dbed0cbe6b8d889baa7f0bd1defd25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(4)=4^3-6\\cdot 4^2+8\\cdot 4 = 64-96+32=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"342\" 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-a055f8dc436c65e9419e66d9af651518_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-4)=(-4)^3-6\\cdot (-4)^2+8\\cdot (-4) = -64-96-32=-192\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"483\" 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-907d1e28e52de3fb47a1c043f30f9ae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(8)=8^3-6\\cdot 8^2+8\\cdot 8 = 512-384+64=192\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"376\" 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-909a39e1a028550d7668da5b6bc4b645_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-8)=(-8)^3-6\\cdot (-8)^2+8\\cdot (-8) = -512-384-64=-960\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"502\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o o polin\u00f4mio desaparece quando <em>x<\/em> \u00e9 +2 ou +4, ent\u00e3o esses valores s\u00e3o as ra\u00edzes do polin\u00f4mio. No entanto, tamb\u00e9m precisamos de adicionar a raiz 0 que encontr\u00e1mos no in\u00edcio do problema. Concluindo, todas as ra\u00edzes do polin\u00f4mio s\u00e3o:<\/p>\n<p class=\"has-text-align-center\"> <strong>Ra\u00edzes ou zeros do polin\u00f4mio<\/strong> : 0, +2 e +4<\/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> Use as propriedades das ra\u00edzes dos polin\u00f4mios para calcular as ra\u00edzes do seguinte polin\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-c4239d0a1f7d60dc0777b21e2358fe0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(x)=(x-1)(x+3)(x^2-x-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"263\" 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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Como vimos na sexta propriedade das ra\u00edzes, quando o polin\u00f4mio \u00e9 formado pelo produto dos fatores, n\u00e3o \u00e9 necess\u00e1rio calcular todas as ra\u00edzes, pois as ra\u00edzes do polin\u00f4mio inteiro s\u00e3o as ra\u00edzes de cada fator.<\/p>\n<p class=\"has-text-align-left\"> Al\u00e9m disso, da segunda propriedade das ra\u00edzes dos polin\u00f4mios, podemos deduzir que a raiz do primeiro fator \u00e9 +1 e a raiz do segundo fator \u00e9 -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-607af42f41a1a5a52051391d5d47ca3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (x-1) \\ \\longrightarrow \\ \\text{ra\\'iz} \\ x=+1\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-791d22dce6bc5be0f80587bcb546aa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle (x+3) \\ \\longrightarrow \\ \\text{ra\\'iz} \\ x=-3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"195\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, s\u00f3 precisamos de determinar as ra\u00edzes do \u00faltimo fator. Para fazer isso, encontramos os divisores do termo independente (-2):<\/p>\n<p class=\"has-text-align-center\"> Divisores de -2: +1, -1, +2, -2<\/p>\n<p class=\"has-text-align-left\"> Portanto, as poss\u00edveis ra\u00edzes ou zeros do \u00faltimo polin\u00f4mio s\u00e3o \u00b11 e \u00b12. Com o qual devemos calcular o valor num\u00e9rico do referido polin\u00f4mio em todos estes valores: <\/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-bbe8195e9f2004a13690e93c706dda7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q(x)= x^2-x-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"135\" 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-99c067ce0e31bf3211cca04bcfce7426_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q(1)=1^2-1-2=1-1-2=-2\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"272\" 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-f1180dacd124c0c5f5b638814d572bcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q(-1)=(-1)^2-(-1)-2=1+1-2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"328\" 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-a685aef3ac803e8600783a6b05555479_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q(2)=2^2-2-2=4-2-2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"259\" 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-0311cf59227e5a6fcd7b92715e080f06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"q(-2)=(-2)^2-(-2)-2=4+2-2=4\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"328\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As ra\u00edzes do polin\u00f4mio \u00e0 direita s\u00e3o, portanto, -1 e 2.<\/p>\n<p class=\"has-text-align-left\"> Portanto, as ra\u00edzes de todo o polin\u00f4mio s\u00e3o todas as ra\u00edzes encontradas:<\/p>\n<p class=\"has-text-align-center\"> <strong>Ra\u00edzes ou zeros do polin\u00f4mio<\/strong> : +1, -1, +2, -3 <\/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 descobrir\u00e1 o que s\u00e3o as ra\u00edzes de um polin\u00f4mio e como elas s\u00e3o calculadas. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo nas ra\u00edzes de um polin\u00f4mio. Quais s\u00e3o as ra\u00edzes de um polin\u00f4mio? Em matem\u00e1tica, as ra\u00edzes (ou zeros) de um polin\u00f4mio s\u00e3o os valores que cancelam &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/raizes-de-um-polinomio\/\"> <span class=\"screen-reader-text\">Ra\u00edzes de um polin\u00f4mio<\/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":[21],"tags":[],"class_list":["post-55","post","type-post","status-publish","format-standard","hentry","category-polinomios"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Como calcular as ra\u00edzes de um polin\u00f4mio? 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