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

1 Answer
Dec 14, 2014

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