Question

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

Answer 1

Factor of Polynomial.

If $(x+a)$ is a factor of a given polynomial $P\left(x\right)$. Then, $P(-a)=0$. (Factor theorem)

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

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

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

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

Hence $x=-1$ is not zero of the polynomial

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