How do I find the partial-fraction decomposition of s+3(s+5)(s2+4s+5)?
1 Answer
Since we have an irreducible denominator the decomposition will look like this:
As+5+Bs+Cs2+4s+5
Now we have to (kind of take a step back and) find a common denominator (the same that we started with):
A(s2+4s+5)+(Bs+C)(s+5)(s+5)(s2+4s+5)=
=(A+B)s2+(4A+5B+C)s+(5A+5C)(s+5)(s2+4s+5)
and since this thingy is the same one that the one is your question we have that
(A+B)s2+(4A+5B+C)s+(5A+5C)=0s2+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=−15
B=15
C=45