To add the given rational expressions, we first need to find a common denominator. The given fractions are:
[tex]\[
\frac{z}{z^2 + 9z + 8} + \frac{8}{z^2 + 12z + 32}
\][/tex]
First, let's factor the denominators.
1. For [tex]\(z^2 + 9z + 8\)[/tex]:
[tex]\[
z^2 + 9z + 8 = (z + 1)(z + 8)
\][/tex]
2. For [tex]\(z^2 + 12z + 32\)[/tex]:
[tex]\[
z^2 + 12z + 32 = (z + 4)(z + 8)
\][/tex]
So, the expressions become:
[tex]\[
\frac{z}{(z + 1)(z + 8)} + \frac{8}{(z + 4)(z + 8)}
\][/tex]
The common denominator for these fractions is the product of all different factors appearing in the denominators:
[tex]\[
(z + 1)(z + 4)(z + 8)
\][/tex]
Next, we need to rewrite each fraction with this common denominator.
1. For the first fraction:
[tex]\[
\frac{z}{(z + 1)(z + 8)} \cdot \frac{z + 4}{z + 4} = \frac{z(z + 4)}{(z + 1)(z + 4)(z + 8)}
= \frac{z^2 + 4z}{(z + 1)(z + 4)(z + 8)}
\][/tex]
2. For the second fraction:
[tex]\[
\frac{8}{(z + 4)(z + 8)} \cdot \frac{z + 1}{z + 1} = \frac{8(z + 1)}{(z + 1)(z + 4)(z + 8)}
= \frac{8z + 8}{(z + 1)(z + 4)(z + 8)}
\][/tex]
Now we can add these fractions together:
[tex]\[
\frac{z^2 + 4z}{(z + 1)(z + 4)(z + 8)} + \frac{8z + 8}{(z + 1)(z + 4)(z + 8)} = \frac{z^2 + 4z + 8z + 8}{(z + 1)(z + 4)(z + 8)}
\][/tex]
Combine like terms in the numerator:
[tex]\[
\frac{z^2 + 12z + 8}{(z + 1)(z + 4)(z + 8)}
\][/tex]
We know (z+1)(z+4)(z+8) expands to [tex]\(z^3 + 13z^2 + 44z + 32\)[/tex]. Therefore, our final expression is:
[tex]\[
\frac{z^2 + 12z + 8}{z^3 + 13z^2 + 44z + 32}
\][/tex]
Thus, the sum of the given rational expressions, simplified, is:
[tex]\[
\boxed{\frac{z^2 + 12z + 8}{z^3 + 13z^2 + 44z + 32}}
\][/tex]
