{"id":203529,"date":"2023-10-28T04:52:05","date_gmt":"2023-10-28T07:52:05","guid":{"rendered":"http:\/\/clubes.obmep.org.br\/blog\/?page_id=203529"},"modified":"2023-10-28T06:17:31","modified_gmt":"2023-10-28T09:17:31","slug":"problema-de-gincana-uma-equacao-e-tres-raizes","status":"publish","type":"page","link":"https:\/\/clubes.obmep.org.br\/blog\/problema-de-gincana-uma-equacao-e-tres-raizes\/","title":{"rendered":".Problema de Gincana: Uma equa\u00e7\u00e3o e tr\u00eas ra\u00edzes"},"content":{"rendered":"

✏ Link do problema para dispositivos da Apple<\/em>.<\/span><\/a><\/strong><\/p>\n

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Problema<\/p>\n

\n A equa\u00e7\u00e3o [tex]\\boxed{x^3+3x^2-2020x+1=0}[\/tex] admite as ra\u00edzes [tex]x_1, \\, x_2[\/tex] e [tex] x_3 .[\/tex]<\/p>\n

Qual o valor de [tex]\\boxed{\\sqrt[3]{x_1}+\\sqrt[3]{x_2}+\\sqrt[3]{x_3}}?[\/tex] <\/strong>\n<\/div>\n

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\"explicador_p\" <\/p>\n

AJUDA<\/p>\n

\n

✏<\/strong> <\/span> As Rela\u00e7\u00f5es de Girard<\/em><\/strong> para uma equa\u00e7\u00e3o c\u00fabica da forma [tex]\\boxed{x^3+kx^2+ lx+m=0}[\/tex] garantem que, se considerarmos que [tex]x_1[\/tex], [tex]x_2[\/tex] e [tex]x_3[\/tex] s\u00e3o as ra\u00edzes dessa equa\u00e7\u00e3o, ent\u00e3o:
\n[tex]\\qquad \\qquad x_1+x_2+x_3=-k .[\/tex]\n<\/div>\n

\n

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

\nInicialmente, observe a seguinte sequ\u00eancia de igualdades equivalentes:
\n[tex]\\qquad x^3+3x^2-2020x+1=0 \\iff x^3+3x^2+3x+1=2023x \\iff\\\\
\n\\qquad (x+1)^3=2023x \\iff \\sqrt[3]{x}=\\dfrac{x+1}{\\sqrt[3]{2023}}.\\\\
\n[\/tex]
\nDessa forma, como <\/p>\n