(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the partial fraction decomposition

1.

[tex]\frac{x^4}{(x-1)^3}[/tex]

2.

[tex]\frac{1}{a^2-x^2}[/tex]

2. Relevant equations

3. The attempt at a solution

1.

[tex]\frac{x^4}{(x-1)^3}[/tex]

[tex]\frac{x^4}{(x-1)^3}=\frac{a}{(x-1)}+\frac{b}{(x-1)^2}+\frac{c}{(x-1)^3}[/tex]

[tex]x^4=(x-1)^2+(x-1)b+c[/tex]

[tex]x^4=ax^2+(-2a+b)x+(a-b+c)[/tex]

I can determine that c is 1 by subbing in 1 for x to get rid of the [tex](x-1)[/tex] terms.

If I plug in a number that dosen't eliminate any terms I get a two variable equation (plugging in my value for c) and I can't solve that to get a numerical answer.

System of equations?

[tex]x^4=ax^2+(-2a+b)x+(a-b+c)[/tex]

all the right sides of the system would be 0 because none of the terms on the right side correspond to the terms on the left side and that would yield a=0 b=0 c=0 which is wrong.

2.

[tex]\frac{1}{a^2-x^2}[/tex]

[tex]\frac{1}{(a+x)(a-x)}[/tex]

[tex]\frac{1}{a^2-x^2}=\frac{a}{(a+x)}+\frac{b}{(a-x)}[/tex]

[tex]1=(a-x)a+(a+x)b[/tex]

From here I'm not quite sure what to do, if you sub in x=a you get

[tex]1=2ab[/tex]

[tex]\frac{1}{2a}=b[/tex]

could you also have

[tex]\frac{1}{2b}=a[/tex]

is x=-a you get [tex]\frac{1}{\sqrt{2}}=a[/tex]

and if you foil and get it into polynomial form...

[tex]1=(-a+b)x+(ba+a^2)[/tex]

If you created a matrix from this you would have a non linear system of equations, I don't know how to solve those and I don't think you need to know how to solve this problem.

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# Homework Help: Partial Fraction Decomposition #3

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