# How do you express (-5s-36) / [ (s+2) (s^2+9) ] in partial fractions?

Nov 5, 2016

The answer is $= - \frac{2}{s + 2} + \frac{2 s - 9}{{s}^{2} + 9}$

#### Explanation:

Let's start the decomposition in partial fractions
$\frac{- 5 s - 36}{\left(s + 2\right) \left({s}^{2} + 9\right)} = \frac{A}{s + 2} + \frac{B s + C}{{s}^{2} + 9}$
$= \frac{A \left({s}^{2} + 9\right) + \left(B s + C\right) \left(s + 2\right)}{\left(s + 2\right) \left({s}^{2} + 9\right)}$

$\therefore - 5 s - 36 = A \left({s}^{2} + 9\right) + \left(B s + C\right) \left(s + 2\right)$
let $s = - 2$$\implies$$- 26 = 13 A$$\implies$A=-2 -36=9A+2C=>$C = - 9$
coeficients of s, $- 5 = 2 B + C$$\implies$$B = 2$
$\therefore \frac{- 5 s - 36}{\left(s + 2\right) \left({s}^{2} + 9\right)} = - \frac{2}{s + 2} + \frac{2 s - 9}{{s}^{2} + 9}$