Question

Determine which of the polynomial has $(x+1)$ a factor: $x^4+x^3+x^2+x+1$

Answer 1

The factor of Polynomial.

If $(x+a)$ is a factor of a given polynomial $P\left(x\right)$. Then, $P(-a)=0$ and $-a$ is called zero of the polynomial

Here,$P\left(x\right)=x^4+x^3+x^2+x+1$

Need to check if $(x+1)$ is a factor of $x^4+x^3+x^2+x+1$ or not.

Substitute $x=-1$ in the given polynomial:

$$\begin{gathered} P(-1)={{(-1)}^4+(-1)}^3+{(-1)}^2+(-1)+1 \\ \Rightarrow P(-1)=1-1+1-1+1 \\ \Rightarrow P(-1)=1\ne 0 \end{gathered}$$

Hence,$x=-1$ is not a zero of the given polynomial.

Hence, $(x+1)$ is not a factor of the given polynomial.