Question

Determine which of the following polynomials has $(x+1)$ a factor :

(ii) $x^4+x^3+x^2+x+1$

Answer 1

STEP 1 : Determining whether polynomial has $(x+1)$ a factor or not

Let $p\left(x\right)=x^4+x^3+x^2+x+1$ be the polynomial

By factor theorem

If $p\left(x\right)$ is a polynomial of degree $n>1$ and $a$ is any real number, then $x–a$ is a factor of $p\left(x\right)$, if $p\left(a\right)=0$,

Therefore, $(x+1)$ is a factor of $p\left(x\right)$, if $p(-1)=0$

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

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

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

$p(-1)=-2+3=1$

Since, $p(-1)\ne 0$

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