# How do I find the partial-fraction decomposition of (s+3)/((s+5)(s^2+4s+5))?

Dec 14, 2014

Since we have an irreducible denominator the decomposition will look like this:

$\frac{A}{s + 5} + \frac{B s + C}{{s}^{2} + 4 s + 5}$

Now we have to (kind of take a step back and) find a common denominator (the same that we started with):

$\frac{A \left({s}^{2} + 4 s + 5\right) + \left(B s + C\right) \left(s + 5\right)}{\left(s + 5\right) \left({s}^{2} + 4 s + 5\right)} =$

$= \frac{\left(A + B\right) {s}^{2} + \left(4 A + 5 B + C\right) s + \left(5 A + 5 C\right)}{\left(s + 5\right) \left({s}^{2} + 4 s + 5\right)}$

and since this thingy is the same one that the one is your question we have that

$\left(A + B\right) {s}^{2} + \left(4 A + 5 B + C\right) s + \left(5 A + 5 C\right) = 0 {s}^{2} + s + 3$

Now two polynomial are equal when coefficients are equal repectively.

So in our case, we have:

$A + B = 0$

$4 A + 5 B + C = 1$

$5 A + 5 C = 3$

Solving this system of equations gives the answer:

$A = - \frac{1}{5}$

$B = \frac{1}{5}$

$C = \frac{4}{5}$