Question

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

Answer 1

To check whether $(x+1)$ is the factor of polynomial $x^4-x^3+x^2-x+1$:

We take, $x+1=0$ $⟹$ $x=-1$.

Replace $x$ by $-1$, we get,

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

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

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

$p(-1)=5$.

That is, value of $p(-1)$ is not zero.

Then $x+1$ is not the factor of polynomial $x^4-x^3+x^2-x+1$.