Solve for: (z-11)/(z^2+11z+30)

Expression: $\frac{ z-11 }{ {z}^{2}+11z+30 }$

To find the excluded values, set the denominator equal to $0$

${z}^{2}+11z+30=0$

Write $11z$ as a sum

${z}^{2}+6z+5z+30=0$

Factor out $z$ from the expression

$z \times \left( z+6 \right)+5z+30=0$

Factor out $5$ from the expression

$z \times \left( z+6 \right)+5\left( z+6 \right)=0$

Factor out $z+6$ from the expression

$\left( z+6 \right) \times \left( z+5 \right)=0$

When the product of factors equals $0$, at least one factor is $0$

$\begin{array} { l }z+6=0,\\z+5=0\end{array}$

Solve the equation for $z$

$\begin{array} { l }z=-6,\\z+5=0\end{array}$

Solve the equation for $z$

$\begin{array} { l }z=-6,\\z=-5\end{array}$

The excluded values of the initial expression are $\begin{array} { l }-6,& -5\end{array}$

$\begin{array} { l }z=-6,& z=-5\end{array}$