Question

Find the value of $p$ for which $(x+2)$ is a factor of the polynomial
$f\left(x\right)=2x^3+p{x}^2+x-10$

Answer 1

Given polynomial is $f\left(x\right)=2x^3+p{x}^2+x-10$.

And $(x+2)$ is a factor of $f\left(x\right)$.

According to the factor theorem, if $(x-a)$ is a factor of $f\left(x\right)$, then $f\left(a\right)=0$.

Hence, $f(-2)=0$

$\therefore 2(-2{)}^3+p(-2{)}^2+(-2)-10=0$

$4p=12+16$

$\therefore p=7$

Hence, if $(x+2)$ is a factor of the given polynomial, $f\left(x\right)$, then $p=7$.