How does one go about decomposing this fraction?

$\displaystyle \frac{2x^2 + 1}{x^3 + 2x^2 + x}$

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- Nov 12th 2009, 11:49 AMSHiFTPartial Fraction Decomposition
How does one go about decomposing this fraction?

$\displaystyle \frac{2x^2 + 1}{x^3 + 2x^2 + x}$ - Nov 12th 2009, 12:09 PMstapel
First,

**take the common factor**in the denominator out front. Then**factor**the remaining quadratic.

Then do the set-up for**partial-fraction decomposition**: Create fractions with the linear factors (with appropriate powers) as the denominators and letters (A, B, C, etc) as numerators. Sum these fractions (but do*not*combine them!), and set this sum equal to the original fraction.

$\displaystyle \frac{A}{x}\, +\, \frac{B}{x\, +\, 1}\, +\, \frac{C}{(x\, +\, 1)^2}$

Then multiply through by the denominator, and solve the resulting equation for the values of the letters. (This is often done by "equating coefficients".)

If you get stuck, please reply showing your steps and reasoning so far, starting with your factorization of the denominator. Thank you!