How do I find the partial-fraction decomposition of (s+3)/((s+5)(s^2+4s+5))?
1 Answer
Since we have an irreducible denominator the decomposition will look like this:
A/(s+5) + (Bs+C)/(s^2+4s+5)
Now we have to (kind of take a step back and) find a common denominator (the same that we started with):
(A(s^2+4s+5)+(Bs+C)(s+5))/((s+5)(s^2+4s+5))=
=((A+B)s^2+(4A+5B+C)s+(5A+5C))/((s+5)(s^2+4s+5))
and since this thingy is the same one that the one is your question we have that
(A+B)s^2+(4A+5B+C)s+(5A+5C)=0s^2+s+3
Now two polynomial are equal when coefficients are equal repectively.
So in our case, we have:
A+B=0
4A+5B+C=1
5A+5C=3
Solving this system of equations gives the answer:
A=-1/5
B=1/5
C=4/5