# How do you write the partial fraction decomposition of the rational expression x^3/[x(x^2+2x+1)]?

Mar 2, 2016

${x}^{3} / \left({x}^{3} + 2 {x}^{2} + x\right) = 1 + \frac{- 2 {x}^{2} - x}{x {\left(x + 1\right)}^{2}}$
$\frac{- 2 {x}^{2} - x}{x {\left(x + 1\right)}^{2}} = \frac{A}{x} + \frac{B}{x + 1} + \frac{C}{x + 1} ^ 2 \to - 2 {x}^{2} - x = A {\left(x + 1\right)}^{2} + B \left(x \left(x + 1\right)\right) + C x \to - 2 {x}^{2} - x = A {x}^{2} + 2 A x + A + B {x}^{2} + B x + C x \to - 2 = A + B , - 1 = 2 A + B + C , A = 0 , B = - 2 , C = 1$
${x}^{3} / \left({x}^{3} + 2 {x}^{2} + x\right) = 1 + \frac{- 2}{x + 1} + \frac{1}{x + 1} ^ 2$

#### Explanation:

First do the long division and divide the denominator to the numerator. Then the remaining fraction we decompose it into partial fractions. Then multiply everything by the common denominator. Finally equate the coefficients of the like terms on both sides and solve for A,B,C then write out your final answer.