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

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

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Answer 1 · 2014-12-14T04:15:23

Since we have an irreducible denominator the decomposition will look like this:

$\frac{A}{s + 5} + \frac{B s + C}{s^{2} + 4 s + 5}$ $A/(s+5) + (Bs+C)/(s^2+4s+5)$

Now we have to (kind of take a step back and) find a common denominator (the same that we started with):

$\frac{A (s^{2} + 4 s + 5) + (B s + C) (s + 5)}{(s + 5) (s^{2} + 4 s + 5)} =$ $(A(s^2+4s+5)+(Bs+C)(s+5))/((s+5)(s^2+4s+5))=$

$= \frac{(A + B) s^{2} + (4 A + 5 B + C) s + (5 A + 5 C)}{(s + 5) (s^{2} + 4 s + 5)}$ $=((A+B)s^2+(4A+5B+C)s+(5A+5C))/((s+5)(s^2+4s+5))$

and since this thingy is the same one that the one is your question we have that

$(A + B) s^{2} + (4 A + 5 B + C) s + (5 A + 5 C) = 0 s^{2} + s + 3$ $(A+B)s^2+(4A+5B+C)s+(5A+5C)=0s^2+s+3$

Now two polynomial are equal when coefficients are equal repectively.

So in our case, we have:

$A + B = 0$ $A+B=0$

$4 A + 5 B + C = 1$ $4A+5B+C=1$

$5 A + 5 C = 3$ $5A+5C=3$

Solving this system of equations gives the answer:

$A = - \frac{1}{5}$ $A=-1/5$

$B = \frac{1}{5}$ $B=1/5$

$C = \frac{4}{5}$ $C=4/5$

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

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How do I find the partial-fraction decomposition of s+3(s+5)(s2+4s+5)(s+3)/((s+5)(s^2+4s+5))?

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