How do you write the partial fraction decomposition of the rational expression (3x^2 + 4x) / (x^2 +1)^2?

Dec 1, 2016

The answer is $= \frac{3}{{x}^{2} + 1} + \frac{4 x - 3}{{x}^{2} + 1} ^ 2$

Explanation:

Let's do the decomposition in partial fractions

$\frac{3 {x}^{2} + 4 x}{{x}^{2} + 1} ^ 2 = \frac{A x + B}{{x}^{2} + 1} + \frac{C x + D}{{x}^{2} + 1} ^ 2$

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

Therefore,

$\left(3 {x}^{2} + 4 x\right) = \left(\left(A x + B\right) \left({x}^{2} + 1\right) + \left(C x + D\right)\right)$

Let $x = 0$, $\implies$, $0 = B + D$

Coefficients of ${x}^{2}$

$3 = B$

Coefficients of $x$

$4 = A + C$

Coefficients of ${x}^{3}$

$0 = A$

$C = 4$

$= B + D$

$D = - 3$

So,

$\frac{3 {x}^{2} + 4 x}{{x}^{2} + 1} ^ 2 = \frac{3}{{x}^{2} + 1} + \frac{4 x - 3}{{x}^{2} + 1} ^ 2$