{"id":116189,"date":"2018-11-26T23:42:55","date_gmt":"2018-11-27T01:42:55","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=116189"},"modified":"2023-05-13T03:58:00","modified_gmt":"2023-05-13T06:58:00","slug":"problemao-equacao-determinante","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problemao-equacao-determinante\/","title":{"rendered":".Problem\u00e3o: Equa\u00e7\u00e3o determinante"},"content":{"rendered":"<p style=\"margin:10px\">\n<div class=\"g1n\" style=\"line-height:35px;\">\n<p class=\"tit\"style=\"line-height:25px;margin:10px\">Problema<br \/>\n<span style=\"font-size:12px;color:#800000\"> (Indicado a partir do 2\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\"\/>\n<p>Seja a fun\u00e7\u00e3o [tex]f: \\mathbb{R} \\to \\mathbb{R}[\/tex] definida por<\/p>\n<p style=\"text-align:center;\">[tex] f(x)=Det \\left[ \\begin{array}{cccc}<br \/>\n1&#038; 2 &#038; 4 &#038; 8 \\\\<br \/>\n1 &#038; 3 &#038; 9 &#038; 27\\\\<br \/>\n1 &#038; 4 &#038; 16 &#038; 64 \\\\<br \/>\n1 &#038; x&#038; x^2 &#038; x^3\\end{array} \\right][\/tex].<\/p>\n<p>Encontre tr\u00eas solu\u00e7\u00f5es para a equa\u00e7\u00e3o [tex]f(x)=0.[\/tex]<\/p>\n<p style=\"margin:20px\">\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 1 <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p>Para encontrar ra\u00edzes da fun\u00e7\u00e3o [tex]f[\/tex], basta usar a propriedade dos determinantes que garante que <\/p>\n<ul>\n<li>Se uma matriz quadrada [tex]A[\/tex] possui duas linhas id\u00eanticas, ent\u00e3o [tex]Det A=0[\/tex].<\/li>\n<\/ul>\n<p>Desta propriedade segue facilmente que <\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad \\qquad f(2)=Det \\left[ \\begin{array}{cccc}<br \/>\n1&#038; 2 &#038; 4 &#038; 8 \\\\<br \/>\n1 &#038; 3 &#038; 9 &#038; 27\\\\<br \/>\n1 &#038; 4 &#038; 16 &#038; 64 \\\\<br \/>\n1 &#038; 2&#038; 4 &#038; 8\\end{array} \\right]=0,[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p>pois a primeira linha e a quarta linha da matriz s\u00e3o id\u00eanticas.<br \/>\nDe forma an\u00e1loga, podemos concluir que [tex]x=3[\/tex] e [tex]x=4[\/tex] tamb\u00e9m s\u00e3o ra\u00edzes:<\/p>\n<p style=\"margin:10px\">\n<p>[tex]\\qquad f(3)=Det \\left[ \\begin{array}{cccc}<br \/>\n1&#038; 2 &#038; 4 &#038; 8 \\\\<br \/>\n1 &#038; 3 &#038; 9 &#038; 27\\\\<br \/>\n1 &#038; 4 &#038; 16 &#038; 64 \\\\<br \/>\n1 &#038; 3 &#038; 9 &#038; 27\\end{array} \\right]=0,<br \/>\n\\qquad f(4)=Det \\left[ \\begin{array}{cccc}<br \/>\n1&#038; 2 &#038; 4 &#038; 8 \\\\<br \/>\n1 &#038; 3 &#038; 9 &#038; 27\\\\<br \/>\n1 &#038; 4 &#038; 16 &#038; 64 \\\\<br \/>\n1 &#038; 4 &#038; 16 &#038; 64\\end{array} \\right]=0,<br \/>\n[\/tex]<\/p>\n<p style=\"margin:10px\">\n<p> Logo, [tex] \\, \\fcolorbox{black}{#eee0e5}{$x=2$} \\, [\/tex], [tex] \\, \\fcolorbox{black}{#eee0e5}{$x=3$} \\, [\/tex] e [tex] \\, \\fcolorbox{black}{#eee0e5}{$x=4$} \\, [\/tex] s\u00e3o tr\u00eas ra\u00edzes da equa\u00e7\u00e3o [tex]f(x)=0.[\/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:30px\">\n<table style=\" height:auto;text-align:justify;margin-left: auto; margin-right: auto; width: 583px;\">\n<tr>\n<td class=\"bf\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1001 alignright\" style=\"border: 0px;\" alt=\"\" src=\"http:\/\/clubes.obmep.org.br\/blog\/wp-content\/uploads\/2013\/01\/menino.jpg\" width=\"95\" height=\"115\" \/><\/td>\n<td class=\"bf\">\n<p class=\"fala\" style=\"width:450px;background-color: #eee0e5;\" > Ao ler a <strong>Solu\u00e7\u00e3o 1<\/strong>, voc\u00ea ficou na d\u00favida se n\u00e3o existem outras solu\u00e7\u00f5es para que o determinante seja zero?<br \/>\nLeia, ent\u00e3o a pr\u00f3xima solu\u00e7\u00e3o!<\/p>\n<\/td>\n<\/tr>\n<\/table>\n<div class=\"g1n\" style=\"line-height: 30px;\">\n<p class=\"tit\" style=\"line-height:30px;margin:10px\">Solu\u00e7\u00e3o 2 <\/p>\n<hr style=\"height:1px; width:100%;background-color:#800000; margin-top: 10px; margin-bottom: 20px;border: 0 none\"\/>\n<p> Observe que a matriz em quest\u00e3o \u00e9 uma <strong> matriz de Vandermonde<\/strong> (veja o problema 2.4 da p\u00e1gina 220 <a href=\"http:\/\/moodle.profmat-sbm.org.br\/MA33\/2012\/AL_PROFMAT_cap08.pdf\" rel=\"noopener\" target=\"_blank\"><strong>deste texto<\/strong><\/a>)<br \/>\nPortanto, seu determinante \u00e9 calculado pelo produto:<br \/>\n[tex]\\qquad \\qquad (x-4) \\cdot (x-3)\\cdot(x-2)\\cdot(4-3)\\cdot(4-2)\\cdot(3-2)[\/tex]<br \/>\nAssim,<br \/>\n[tex]\\qquad \\qquad f(x)=(x-4) \\cdot (x-3)\\cdot(x-2)\\cdot1\\cdot2\\cdot1[\/tex]<br \/>\nDessa forma, para encontrar as solu\u00e7\u00f5es da equa\u00e7\u00e3o [tex]f(x)=0[\/tex], basta fazer:<br \/>\n[tex]\\qquad \\qquad (x-4) \\cdot (x-3)\\cdot(x-2)\\cdot1\\cdot2\\cdot1=0[\/tex]<br \/>\nPortanto, as solu\u00e7\u00f5es para a equa\u00e7\u00e3o [tex]f(x)=0[\/tex] s\u00e3o dadas por:<\/p>\n<ul>\n<li>[tex]x-4=0 [\/tex], donde [tex]x=4[\/tex];<\/li>\n<li>[tex]x-3=0 [\/tex], donde [tex]x=3[\/tex];<\/li>\n<li>[tex]x-2=0 [\/tex], donde [tex]x=2[\/tex].<\/li>\n<\/ul>\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 2\u00ba ano do E. M.) Seja a fun\u00e7\u00e3o [tex]f: \\mathbb{R} \\to \\mathbb{R}[\/tex] definida por [tex] f(x)=Det \\left[ \\begin{array}{cccc} 1&#038; 2 &#038; 4 &#038; 8 \\\\ 1 &#038; 3 &#038; 9 &#038; 27\\\\ 1 &#038; 4 &#038; 16 &#038; 64 \\\\ 1 &#038; x&#038; x^2 &#038; x^3\\end{array} \\right][\/tex]. Encontre tr\u00eas solu\u00e7\u00f5es &hellip; <\/p>\n<p><a class=\"more-link btn\" href=\"https:\/\/clubes.obmep.org.br\/blog\/problemao-equacao-determinante\/\">Continue lendo<\/a><\/p>\n","protected":false},"author":667,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-116189","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/116189","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\/667"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=116189"}],"version-history":[{"count":21,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/116189\/revisions"}],"predecessor-version":[{"id":189959,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/116189\/revisions\/189959"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=116189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}