Question

Determine whether the following polynomial has $(x+1)$ as a factor:
$x^4+2x^3+2x^2+x+1$.

Answer 1

Given polynomial is $x^4+2x^3+2x^2+x+1$.

To find out: Whether $(x+1)$ is a factor of the given polynomial or not.

Let the given polynomial be $p\left(x\right)$.

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

Hence, $(x+1)$ will be a factor of the given polynomial if $p(-1)=0$.

Let's check for that:

Substituting $x$ as $-1$, we get,

$p\left(x\right)=x^4+2x^3+2x^2+x+1$

$p(-1)=(-1{)}^4+2(-1{)}^3+2(-1{)}^2+(-1)+1$

$p(-1)=1-2+2-1+1$

$p(-1)=1$.

$\therefore p(-1)\ne 0$

Hence, $(x+1)$ is not a factor of the polynomial $x^4+2x^3+2x^2+x+1$.