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

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