How do I find the partial-fraction decomposition of s+3(s+5)(s2+4s+5)?

1 Answer
Dec 14, 2014

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