# How do you express (2x^3 -x^2)/((x^2 +1)^2) in partial fractions?

Feb 14, 2016

Partial fractions are $\frac{2 x + 1}{{x}^{2} + 1} - \frac{2 x + 1}{{x}^{2} + 1} ^ 2$

#### Explanation:

Let the function $\frac{2 {x}^{3} - {x}^{2}}{{x}^{2} + 1} ^ 2$ be written in partial fractions as

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

Solving this becomes $\frac{\left(A x + B\right) \left({x}^{2} + 1\right) + C x + D}{{x}^{2} + 1} ^ 2$

As denominator in given function is same

it follows that

$\left(A x + B\right) \left({x}^{2} + 1\right) + C x + D \Leftrightarrow \left(2 {x}^{3} - {x}^{2}\right)$ or

$A {x}^{3} + B {x}^{2} + \left(A + C\right) x + \left(B + D\right) \Leftrightarrow \left(2 {x}^{3} - {x}^{2}\right)$

Comparing like terms

$A = 2 , B = - 1 , A + C = 0 \mathmr{and} B + D = 0$ i.e.

$A = 2 , B = - 1 , C = - 2 \mathmr{and} D = - 1$

Hence partial fractions are $\frac{2 x + 1}{{x}^{2} + 1} - \frac{2 x + 1}{{x}^{2} + 1} ^ 2$