Question

If $(x-2)$ is a factor of $x^3+a{x}^2+bx+16$ and $b=4a$, find the value of $a$ and $b$.

Answer 1

If $(x-2)$ is a factor then $2$ is a root of polynomial.

$x^3+a{x}^2+bx+16$

$(2{)}^3+a(2{)}^2+b\left(2\right)+16=0$

$8+4a+2b+16=0$

$4a+2b+24=0$

$2a+b+12=0$ -----(i)

Given $b=4a$

$2a+4a+12=0$

$6a+12=0$

$a=-2$ and $b=4(-2)$

$a=-2;b=-8$