Question

Si las raices de la ecuacion :

x³ + 2x²+2=0
son a,b;c, calcular:

P(a/bc + b/ac + c/ab)

si:p(x) = x^2004 + 2x^2003 - 5

A) 6 B)5 C) -5 D) 10 E)15

Answer 1

$(1) ~~~~~~ (x-a)(x-b)(x-c)= x^3+2x^2+2\\\\ ~~~~~~~ x^3-(ab+ac+bc)x^2-(a+b+c)x-abc= x^3+2x^2+2\\ \\ ~~~~~~~ ab+ac+bc=-2\\ ~~~~~~~ a+b+c=0\\ ~~~~~~~ abc=-2\\ \\ \texttt{Luego necesitaremos...}\\ \\ ~~~~~~~(a+b+c)^2=0\\ ~~~~~~~a^2+b^2+c^2+2(ab+ac+bc)=0\\ ~~~~~~~a^2+b^2+c^2+2(-2)=0\\ \\ ~~~~~~~a^2+b^2+c^2=4\\ \\ ~~~~~~~\dfrac{a^2+b^2+c^2}{abc}=\dfrac{4}{abc}\\ \\ \\ ~~~~~~~\dfrac{a}{bc}+\dfrac{b}{ac}+\dfrac{c}{ab}=\dfrac{4}{-2}\\ \\ \\ ~~~~~~~\dfrac{a}{bc}+\dfrac{b}{ac}+\dfrac{c}{ab}=-2$

$\texttt{por fin:}\\ P(-2)=(-2)^{2004}+2(-2)^{2003}-5\\ \\ P(-2)=2^{2004}-2(2)^{2003}-5\\ \\ P(-2)=2^{2004}-2^{2004}-5\\ \\ \boxed{P(-2)=-5}$