How do you express (x^2 + 33)/(x^3 + x^2) in partial fractions?

Mar 8, 2016

$\frac{{x}^{2} + 33}{{x}^{2} \left(x + 1\right)} = - \frac{33}{x} + \frac{33}{x} ^ 2 + \frac{34}{x + 1}$

Explanation:

$\frac{{x}^{2} + 33}{{x}^{2} \left(x + 1\right)} = \frac{A}{x} + \frac{B}{x} ^ 2 + \frac{C}{x + 1}$

${x}^{2} + 33 = A \left(x \left(x + 1\right)\right) + B \left(x + 1\right) + C {x}^{2}$

${x}^{2} + 33 = A {x}^{2} + A x + B x + B + C {x}^{2}$

$1 = A + C , 0 = A + B , 33 = B$

$A = - 33 , B = 33 , C = 34$

$\frac{{x}^{2} + 33}{{x}^{2} \left(x + 1\right)} = - \frac{33}{x} + \frac{33}{x} ^ 2 + \frac{34}{x + 1}$