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I would like to use partial fraction decomposition on: $\frac{x+1}{x^4+x}$. The denominator can be broken into $x(x^3+1)$. I'm just not sure how to deal with the $x^3+1$ part. Do I put something like: $Ax^2+Bx+C$ on top? That would lead to a step looking like this: $\frac{Ax^2+Bx+C}{x^3+1}+\frac{D}{x}$. Or, is there something else I need to do?

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  • $\begingroup$ $x^3+1$ is divisible by $x-1$ $\endgroup$ – Tito Eliatron Jan 14 at 20:51
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    $\begingroup$ @TitoEliatron, I think you meant $x+1$. $\endgroup$ – RobPratt Jan 14 at 20:54
  • $\begingroup$ Yeah, it was a misprint. Thanks. $\endgroup$ – Tito Eliatron Jan 14 at 20:59
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$$ \frac{x+1}{x^4+x}=\frac{x+1}{x(x+1)(x^2-x+1)}=\frac{1}{x(x^2-x+1)}=\frac{1}{x}-\frac{x+1}{x^2-x+1} $$

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