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

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Answer 1 · 2014-12-14T04:15:23

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:

$A=-1/5$

$B=1/5$

$C=4/5$

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