{"id":82461,"date":"2017-08-23T13:44:01","date_gmt":"2017-08-23T16:44:01","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=82461"},"modified":"2023-04-08T02:35:07","modified_gmt":"2023-04-08T05:35:07","slug":"problemao-raizes-de-uma-equacao","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-raizes-de-uma-equacao\/","title":{"rendered":".Problem\u00e3o:  Ra\u00edzes de uma equa\u00e7\u00e3o"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 3\u00ba ano do E. M.)<\/span><\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nSabendo que [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] s\u00e3o as ra\u00edzes da equa\u00e7\u00e3o<br \/>\n[tex]\\qquad \\qquad x^3 + 5x &#8211; 8 =0[\/tex],<br \/>\ndetermine o valor de [tex]x_1^3 + x_2^3 + x_3^3[\/tex].<\/p>\n<p style=\"margin:10px\">\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"cb36 g1\" style=\"margin:30px;line-height:30px; width:85%; margin-left:auto; margin-right:auto; \"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" style=\"border:none\"  alt=\"explicador_p\" width=\"80\" height=\"91\" class=\"alignleft size-full wp-image-1001\" \/> <\/p>\n<p class=\"c3 ti3\" style=\"line-height:30px;margin:10px\"> Lembrete <\/p>\n<p style=\"margin:20px\">\n<p>Uma das rela\u00e7\u00f5es de Girard para polin\u00f4mios de grau [tex]n[\/tex] nos assegura que a soma [tex]S[\/tex] das ra\u00edzes do polin\u00f4mio [tex]\\boxed{a_nx^n+a_{n-1}x^{n-1}+\\ldots+a_1x+a_0}[\/tex], [tex]a_n\\neq0[\/tex], \u00e9 dada por [tex]\\boxed{S=-\\dfrac{a_{n-1}}{a_n}}[\/tex].<\/p>\n<p style=\"font-size:11px; text-align:right;\">\n(Para conhecer um pouco sobre as <strong>rela\u00e7\u00f5es de Girard para equa\u00e7\u00f5es do segundo grau, cliquem <strong><a href=\"http:\/\/clubes.obmep.org.br\/blog\/texto_004-relacoes-de-girard\/\" target=\"_blank\" title=\"Sala de leitura sobre Rela\u00e7\u00f5es de Girard\" >AQUI<\/a><\/strong>)<\/p>\n<\/div>\n<p style=\"margin:30px\">\n<div class=\"g1n\" style=\"line-height:30px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\nSendo [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] as ra\u00edzes da equa\u00e7\u00e3o [tex]x^3 + 5x &#8211; 8 =0[\/tex], temos:<\/p>\n<p style=\"margin:10px\">\n<ul>\n<li>[tex] \\, x_1^3 + 5x_1 &#8211; 8 =0[\/tex],<br \/>\ndonde<br \/>\n[tex]\\fcolorbox{#800000}{}{$x_1^3 = &#8211; 5x_1 + 8$} \\, . \\qquad \\color{#800000}(i)[\/tex]<\/li>\n<li>[tex] \\, x_2^3 + 5x_2 &#8211; 8 =0[\/tex],<br \/>\ndonde<br \/>\n[tex]\\fcolorbox{#800000}{}{$x_2^3 = &#8211; 5x_2 + 8$} \\, . \\qquad \\color{#800000}(ii)[\/tex]<\/li>\n<li>[tex] \\, x_3^3 + 5x_3 &#8211; 8 =0[\/tex],<br \/>\ndonde<br \/>\n[tex]\\fcolorbox{#800000}{}{$x_3^3 = &#8211; 5x_3 + 8$} \\, .\\qquad \\color{#800000}(iii)[\/tex]<\/li>\n<\/ul>\n<p style=\"margin:10px\">\n<p>Somando membro a membro as igualdades  [tex]\\color{#800000}(i)[\/tex], [tex]\\color{#800000}(ii)[\/tex] e [tex]\\color{#800000}(iii)[\/tex], segue que:<br \/>\n[tex]\\qquad x_1^3 + x_2^3 + x_3^3 = -5 (x_1 + x_2 + x_3) + 24[\/tex].<br \/>\nObserve que a equa\u00e7\u00e3o do problema pode ser reescrita como [tex] \\, x^3 +0x^2+ 5x &#8211; 8 =0[\/tex]; assim, das rela\u00e7\u00f5es de <em>Girard<\/em>, temos que<br \/>\n[tex]\\qquad x_1 + x_2 + x_3 = \\dfrac{0}{1}=0[\/tex],<br \/>\ne, portanto,<br \/>\n[tex]\\qquad x_1^3 + x_2^3 + x_3^3 = -5 \\cdot 0 + 24[\/tex].<\/p>\n<p style=\"margin:10px\">\n<p>Assim,  [tex]\\fcolorbox{black}{#eee0e5}{$x_1^3 + x_2^3 + x_3^3 =  24$}[\/tex].<\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 15px;border: 0 none\"\/>\n<p class=\"g2\" style=\"line-height:20px;\">Solu\u00e7\u00e3o elaborada pelos <strong>Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n<p style=\"margin:10px\">\n","protected":false},"excerpt":{"rendered":"<p>Problema (Indicado a partir do 3\u00ba ano do E. M.) Sabendo que [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] s\u00e3o as ra\u00edzes da equa\u00e7\u00e3o [tex]\\qquad \\qquad x^3 + 5x &#8211; 8 =0[\/tex], determine o valor de [tex]x_1^3 + x_2^3 + x_3^3[\/tex]. Lembrete Uma das rela\u00e7\u00f5es de Girard para polin\u00f4mios de grau [tex]n[\/tex] nos assegura que a soma [tex]S[\/tex] &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-raizes-de-uma-equacao\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":179,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-82461","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/82461","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/users\/179"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=82461"}],"version-history":[{"count":40,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/82461\/revisions"}],"predecessor-version":[{"id":186544,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/82461\/revisions\/186544"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=82461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}