# 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#