Question

Sabendo que p(-2)= -10 e que -1 é raiz de P(x)= x3+8x2-cx+d, encontre c e d

Answer 1

EAE manoo,

se P(-2) é -10, e -1 é raiz  de P(x) = x³+8x²-cx+d, então fazemos..

$\begin{cases}P(-2)=-10~\Rightarrow~(-2)^3+8\cdot(-2)^2-c\cdot(-2)+d=-10~~(i)\\ (-1)~e\´~raiz~\Rightarrow~(-1)^3+8\cdot(-1)^2-c\cdot(-1)+d=0~~(ii)\end{cases}\\\\\\ \begin{cases}-8+32+2c+d=-10~~(i)\\ -1+8+c+d=0~~(ii)\end{cases}\\\\\\ \begin{cases}2c+d=-34~~(i)~~(\times~por~-1~e~soma~i~com~ii)\\ c+d=-7~~(ii)\end{cases}\\\\\\ \begin{cases}-2c-d=34~~(i)\\ c+d=-7~~(ii)\end{cases}$

$\begin{cases}-2c-d=34~~(i)\\ ~~~c+d=-7~~(ii)\end{cases}\\ ~~~------\\ ~~~~-c~~~~~=27\\\\ ~~~~~~~~\huge\boxed{c=-27}$

Se c vale -27, então d valerá..

$c+d=-7\\ -27+d=-7\\ d=-7+27\\\\ \huge\boxed{d=20}$

Tenha ótimos estudos ;D