Partial fraction decomposition when one of the partial fraction coefficients is zero

Question

It is quite some time since I used to do partial fraction (PF) decomposition. I am not able to recall how to PF the following expression? $\frac{9s+9}{(s+1)(s+3)}$.

The coefficient of $\frac{1}{s+1}$ comes out to be zero.

Answer 1

If you want PF of $\frac{ax+b}{(cx+d)(px+q)}$, just write it in the form $\frac{m}{(cx+d)}$ + $\frac{n}{(px+q)}$. Then compare both equations to obtain m and n.
In your question, m=0 and n=9; so PF is $\frac{0}{(s+1)}$+$\frac{9}{s+3}$ or simply $\frac{9}{s+3}$; which you would have easily got by cancelling out (s+1) from numerator and denominator.
There's no harm in getting either of m or n as 0;as you say about coefficient of $\frac{1}{s+1}$.