To determine where the given expression is undefined, we need to focus on the denominator. The expression becomes undefined whenever the denominator is equal to zero, because division by zero is undefined.
Given the expression:
[tex]\[
\frac{z^2 + 16z + 64}{z^2 - 3z - 18}
\][/tex]
we focus on the denominator [tex]\( z^2 - 3z - 18 \)[/tex].
We need to solve the equation:
[tex]\[
z^2 - 3z - 18 = 0
\][/tex]
This is a quadratic equation. To solve it, we can factor it into the form [tex]\((z - a)(z - b)\)[/tex]. We need to find two numbers that multiply to [tex]\(-18\)[/tex] and add up to [tex]\(-3\)[/tex].
After testing possible factor pairs of [tex]\(-18\)[/tex]:
[tex]\[
-6 \times 3 = -18 \quad \text{and} \quad -6 + 3 = -3
\][/tex]
So the equation can be factored as:
[tex]\[
(z - 6)(z + 3) = 0
\][/tex]
Setting each factor equal to zero gives us the solutions for [tex]\( z \)[/tex]:
[tex]\[
z - 6 = 0 \quad \Rightarrow \quad z = 6
\][/tex]
and
[tex]\[
z + 3 = 0 \quad \Rightarrow \quad z = -3
\][/tex]
Therefore, the expression is undefined for the values [tex]\( z = 6 \)[/tex] and [tex]\( z = -3 \)[/tex].
[tex]\[
z = 6, -3
\][/tex]
