Question

In the following case, use factor theorem to find whether $g\left(x\right)$ is a factor of the polynomial $p\left(x\right)$ or not.
$p\left(x\right)=2x^4+x^3+4x^2-x-7g\left(x\right)=x+2$

Answer 1

The Factor Theorem states that if $a$ is the root of any polynomial $p\left(x\right)$ that is if $p\left(a\right)=0$, then $(x-a)$ is the factor of the polynomial $p\left(x\right)$.

It is given that $p\left(x\right)=2x^4+x^3+4x^2-x-7$ and $g\left(x\right)=x+2$, therefore, by factor theorem $(x+2)$ is the factor of $p\left(x\right)$ if $p(-2)=0$ and thus,

$p(-2)=(2\times (-2{)}^4)+(-2{)}^3+(4\times (-2{)}^2)-(-2)-7=(2\times 16)-8+(4\times 4)+2-7$

$=32-8+16+2-7=32+16+2-8-7=35\ne 0$

Since $p(-2)\ne 0$

Hence, $(x+2)$ is not a factor of $p\left(x\right)=2x^4+x^3+4x^2-x-7$.