Question

Determine whether $(x+2)$ is a factor of $2x^4+x^3+4x^2-x-7$.

Answer 1

Given, $p\left(x\right)=2x^4+x^3+4x^2-x-7$.

By Remainder Theorem, we know,

if $p\left(x\right)=2x^4+x^3+4x^2-x-7$ is divided by $x+2$, then the remainder obtained is $p(-2)$.

That is,

$p(-2)=2(-2{)}^4+(-2{)}^3+4(-2{)}^2-(-2)-7$

$=32-8+16+2-7$

$=50-15=35\ne 0$.

Since, $p\left(2\right)\ne 0$, we can say that $x+2$ is not a factor of $2x^4+x^3+4x^2-x-7$.