Q7. - Algebra 1

Question

Simplify each expression. State the excluded values of the variables.

$\frac{z-6}{z^2-3z-18}$

Step-by-Step Solution

Answer

The given expression can be simplified to $\frac{1}{z+3}$. Excluded values of the variable are $6$ and $-3$.

Step 1. Factor.

Factor the given expression.

$\begin{array}{c}\frac{\left(z-6\right)}{z^2-3z-18}=\frac{\left(z-6\right)}{\left(z-6\right)\left(z+3\right)}\\ =\frac{1}{z+3}\end{array}$

Step 2. Exclude the values.

Exclude the values for which $z^2-3z-18$ equals 0.

$\begin{array}{c} z^2-3z-18=0\\ \left(z-6\right)\left(z+3\right)=0\end{array}$

Step 3. Zero product property.

Apply zero product property to above equation.

$$\begin{gathered} z-6=0\text{ or }z+3=0 \\ z=6z=-3 \end{gathered}$$

Therefore, $z\ne 6$ and $z\ne -3$.