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posted by  ceverett24 on 11/6/2009 4:16:06 PM  |  status: Closed  |  Earned Karma: 25

Partial fraction decomposition

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N/A N/A N/A N/A 11/7/2009 at 6:00:00 PM
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How would I do partial fraction decomposition on


I have looked on the internet and have not found anything resembling this situation
Tags: Calculus
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posted by hado on 11/6/2009 4:43:32 PM  |  status: Live
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posted by ceverett24 on 11/6/2009 4:50:24 PM  |  status: Live
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oh, i guess i did it right. thanks!
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posted by Ogivol on 11/6/2009 5:06:34 PM  |  status: Live
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Response Details:
This problem can be approached as follows:

You have already divided your denominator into its basic components: and

Set up an equation as follows:


Where a and b are to arbitrary constants you will have to solve for in order to complete the decomposition.
Multiply both sides by the common denominator  in this case :

You are now left with:


At this point one method would be to expand and foil.

Leaving you with: 

Combine like terms to get :
Using a basic understanding of polynomials, you know that in order for two polynomials to be equal, their coefficient must be equal as well. We can get two equations from this fact.

The first and most obvious one would be that : you get this by trying s=0 in the original equation.
The secondary one is slightly harder to see. As there is no term on the opposite side, we can imagine there is one with a coefficient of 0.



You are now left with two equations:



You have a system of equations which can easily be solved for both a and b.
Once a and b are found, plug them into your original decomposition which was :



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