# How do you simplify (z^2-z-6)/(z-6) *( z^2-6z)/(z^2+2z-15)?

Jul 13, 2015

$\frac{{z}^{2} - z - 6}{z - 6} \cdot \frac{{z}^{2} - 6 z}{{z}^{2} + 2 z - 15} = \frac{{z}^{2} + 2 z}{z + 5}$

#### Explanation:

$\frac{{z}^{2} - z - 6}{z - 6} \cdot \frac{{z}^{2} - 6 z}{{z}^{2} + 2 z - 15}$
$\textcolor{w h i t e}{\text{XXXX}}$Combining the numerator factors and the denominator factors:
$= \frac{\left({z}^{2} - z - 6\right) \left({z}^{2} - 6 z\right)}{\left(z - 6\right) \left({z}^{2} + 2 z - 15\right)}$
$\textcolor{w h i t e}{\text{XXXX}}$Factoring
$= \frac{\left(z - 3\right) \left(z + 2\right) \left(z\right) \left(z - 6\right)}{\left(z - 6\right) \left(z + 5\right) \left(z - 3\right)}$
$\textcolor{w h i t e}{\text{XXXX}}$Cancelling out matching terms in the numerator and denominator
$= \frac{\cancel{\left(z - 3\right)} \left(z + 2\right) \left(z\right) \cancel{\left(z - 6\right)}}{\cancel{\left(z - 6\right)} \left(z + 5\right) \cancel{\left(z - 3\right)}}$
$\textcolor{w h i t e}{\text{XXXX}}$Rewrite simplified:
$= \frac{{z}^{2} + 2 z}{z + 5}$