Chapter 8.5, Problem 12WE

To determine

To find: if the binomial is a factor of the provided polynomial

Answer to Problem 12WE

The binomial $z+2$ is a factor of the provided polynomial $P\left(z\right)=z^5+2z^4+z^3+2z^2+z+2$

Explanation of Solution

Given information: the polynomial and binomial provided in the question,

  $z+2;P\left(z\right)=z^5+2z^4+z^3+2z^2+z+2$

Calculation:

Use the factor theorem, $P\left(-2\right)=0$ ,

  $\begin{array}{l}P\left(-2\right)={\left(-2\right)}^5+2{\left(-2\right)}^4+{\left(-2\right)}^3+2{\left(-2\right)}^2+\left(-2\right)+2\\ =-32+32-8+8-2+2\\ =0\end{array}$

Hence, the binomial $z+2$ is a factor of the provided polynomial $P\left(z\right)=z^5+2z^4+z^3+2z^2+z+2$