Chapter 11.6, Problem 18E

To determine

To calculate:

The factor the polynomial:

  $z^3+5z^2-9z-45$

Answer to Problem 18E

The factor of the polynomial of $z^3+5z^2-9z-45$ is $\left(z+5\right)\left(z+3\right)\left(z-3\right)$ .

Explanation of Solution

Given information:

  $z^3+5z^2-9z-45$

The given function is:

  $z^3+5z^2-9z-45$

Factor the function:

  $\left(z^3+5z^2\right)+\left(-9z-45\right)$

Factor out $z^2$ from $z^2+5z^2$ and $-9$ and $-9\left(z+5\right)$ :

  $\begin{array}{l}=\left(z^3+5z^2\right)+\left(-9z-45\right)\\ =\left(z{z}^2+5z^2\right)+\left(-9z-9\cdot 5\right)\left[\because a^{b+c}=a^b{a}^c\right]\end{array}$

Factor the common term $z^2$ and $-9$ :

  $-9\left(z+5\right)+z^2\left(z+5\right)$

Factor the common term $z+5$

  $\left(z+5\right)\left(z^2-9\right)$

Apply the formula

  $\left(z+5\right)\left(z+3\right)\left(z-3\right)\left[\because a^2-b^2=\left(a+b\right)\left(a-b\right)\right]$

The factor of the polynomial is:

  $\left(z+5\right)\left(z+3\right)\left(z-3\right)$