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The question is following:

Write out the form of the partial fraction decomposition of the function (as in this example). Do not determine the numerical values of the coefficients. $$\frac{1}{x^2+x^4}$$

I know how to solve this. but the problem is I have to do it like the example and I can't figure out what the "proper way" the Webassign is looking for.

$$\frac{x^3+x^2+1}{x(x-1)(x^2+x+1)(x^2+1)^3}=\frac{A}{x}+\frac{B}{x-1}+\frac{Cx+D}{x^2+x+1}+\frac{Ex+F}{x^2+1}+\frac{Gx+H}{(x^2+1)^2}+\frac{Ix+J}{(x^2+1)^3}$$

OneMoreGamble
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2 Answers2

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Hint.

Your fraction is equal to $$ \frac{1}{x^2(x^2+1)} $$ and $x^2+1$ is an irreducible quadratic factor.

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Is it $$\frac A {x^2}+\frac B{x^2+1}$$?

No, it's

$$\frac A{x}+\frac B {x^2}+\frac C{x^2+1}$$

Vons
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  • Thanks, I forgot factoring x^2 to x and x^2. I'm more stupid than this question – OneMoreGamble Feb 16 '21 at 00:21
  • $$ \begin{array}{l} \text { Let } y=x^{2} \text {, then } \ \displaystyle \frac{1}{x^{2}+x^{4}}=\frac{1}{y(y+1)}=\frac{1}{y}-\frac{1}{y+1}=\frac{1}{x^{2}}-\frac{1}{x^{2}+1} \end{array} $$ – Lai Nov 21 '21 at 02:07