{"id":36233,"date":"2015-05-01T21:06:20","date_gmt":"2015-05-02T00:06:20","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=36233"},"modified":"2023-03-21T22:02:58","modified_gmt":"2023-03-22T01:02:58","slug":"problema-qual-o-valor-da-expressao","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-qual-o-valor-da-expressao\/","title":{"rendered":".Problema: Qual o valor da express\u00e3o?"},"content":{"rendered":"

\n

\n

Problema
\n (Indicado a partir do 9\u00ba ano do E. F.)<\/span><\/p>\n

\nSeja [tex]x[\/tex] um n\u00famero que satisfaz a equa\u00e7\u00e3o [tex]x^2+x-1=0[\/tex].
\nDetermine o valor da express\u00e3o [tex]x^8-7x^4+1[\/tex].\n<\/div>\n

\n

\n

Solu\u00e7\u00e3o<\/p>\n

\nObserve, inicialmente, que [tex]x\\ne 0[\/tex]; logo, temos que:<\/p>\n

\n[tex]\\quad x^2+x-1=0 \\Leftrightarrow x-\\dfrac{1}{x}=-1 \\Rightarrow \\left(x-\\dfrac{1}{x}\\right)^2=(-1)^2 \\Leftrightarrow x^2-2+\\dfrac{1}{x^2}=1 \\Leftrightarrow[\/tex]
\n[tex]\\quad x^2+\\dfrac{1}{x^2}=3 \\Rightarrow \\left(x^2+\\dfrac{1}{x^2}\\right)^2=(3)^2 \\Leftrightarrow x^4+2+ \\dfrac{1}{x^4}=9 \\Leftrightarrow x^4+\\dfrac{1}{x^4}=7 \\Leftrightarrow[\/tex]
\n[tex]\\quad x^4+\\dfrac{1}{x^4}=7 \\Leftrightarrow x^8-7x^4+1=0[\/tex].<\/p>\n

Assim, se [tex]x^2+x-1=0[\/tex], ent\u00e3o [tex] x^8-7x^4+1=0[\/tex].<\/p>\n

\n

Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n

\n

\nParticipou da discuss\u00e3o o Clube Fermatianos<\/strong>. <\/div>\n","protected":false},"excerpt":{"rendered":"

Problema (Indicado a partir do 9\u00ba ano do E. F.) Seja [tex]x[\/tex] um n\u00famero que satisfaz a equa\u00e7\u00e3o [tex]x^2+x-1=0[\/tex]. Determine o valor da express\u00e3o [tex]x^8-7x^4+1[\/tex]. Solu\u00e7\u00e3o Observe, inicialmente, que [tex]x\\ne 0[\/tex]; logo, temos que: [tex]\\quad x^2+x-1=0 \\Leftrightarrow x-\\dfrac{1}{x}=-1 \\Rightarrow \\left(x-\\dfrac{1}{x}\\right)^2=(-1)^2 \\Leftrightarrow x^2-2+\\dfrac{1}{x^2}=1 \\Leftrightarrow[\/tex] [tex]\\quad x^2+\\dfrac{1}{x^2}=3 \\Rightarrow \\left(x^2+\\dfrac{1}{x^2}\\right)^2=(3)^2 \\Leftrightarrow x^4+2+ \\dfrac{1}{x^4}=9 \\Leftrightarrow x^4+\\dfrac{1}{x^4}=7 \\Leftrightarrow[\/tex] [tex]\\quad x^4+\\dfrac{1}{x^4}=7 … <\/p>\n

Continue lendo<\/a><\/p>\n","protected":false},"author":164,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"closed","template":"template-onecolumn.php","meta":{"footnotes":""},"class_list":["post-36233","page","type-page","status-publish","hentry","nodate","item-wrap"],"_links":{"self":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36233","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\/164"}],"replies":[{"embeddable":true,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/comments?post=36233"}],"version-history":[{"count":20,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36233\/revisions"}],"predecessor-version":[{"id":185091,"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/pages\/36233\/revisions\/185091"}],"wp:attachment":[{"href":"https:\/\/clubes.obmep.org.br\/blog\/wp-json\/wp\/v2\/media?parent=36233"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}