partial fraction decomposition

**jeneverboy** · April 8th 2010, 05:22 AM

Hi

can someone help me with the following partial fraction decomposition. It is important that the answer has no complex number.

$ \frac{4}{s^{2}(s^{2}+1)} = \frac{A}{s^{2}} + \frac{B}{s^{2}+1} $
is
$ A(s^{2}+1) + Bs^{2} = 4 $

**drumist** · April 8th 2010, 05:32 AM

Remember that when a factor has a multiplicity greater than one, you should list that factor along with all its powers. In other words, since $s^2 = s \cdot s$ , we should have factors of $s$ and $s^2$ .

Also, since $s^2+1$ cannot be factored any further, but has degree 2, we need to use two constants in the numerator for that term.

The whole thing should look like this:

$\frac{4}{s^{2}(s^{2}+1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{Cs+D}{s^2+1}$

By the way, a good check to make sure you didn't forget any terms is to make sure the number of constants you have should equal the degree of the original denominator.

**jeneverboy** · April 8th 2010, 05:58 AM

Allright thanks.

$ 4 = As(s^{2}+1) + B(s^{2}+1) + (Cs+D)s^{2} $
4 = A + B
0 = A
0 = B + D
0 = A + C

so
A=0, B=4, D=-4 and C=0

which seems to be correct.

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