# How do you express (x^3 - 5x + 2) / (x^2 - 8x + 15) in partial fractions?

Aug 7, 2016

$\frac{{x}^{3} - 5 x + 2}{{x}^{2} - 8 x + 15} = x + 8 - \frac{7}{x - 3} + \frac{51}{x - 5}$

#### Explanation:

$\frac{{x}^{3} - 5 x + 2}{{x}^{2} - 8 x + 15}$

$= \frac{{x}^{3} - 8 {x}^{2} + 15 x + 8 {x}^{2} - 64 x + 120 + 44 x - 118}{{x}^{2} - 8 x + 15}$

$= \frac{x \left({x}^{2} - 8 x + 15\right) + 8 \left({x}^{2} - 8 x + 15\right) + 44 x - 118}{{x}^{2} - 8 x + 15}$

$= x + 8 + \frac{44 x - 118}{{x}^{2} - 8 x + 15}$

$= x + 8 + \frac{44 x - 118}{\left(x - 3\right) \left(x - 5\right)}$

$= x + 8 + \frac{A}{x - 3} + \frac{B}{x - 5}$

Using Heaviside's cover-up method, we find:

$A = \frac{44 \left(3\right) - 118}{\left(3\right) - 5} = \frac{132 - 118}{- 2} = \frac{14}{- 2} = - 7$

$B = \frac{44 \left(5\right) - 118}{\left(5\right) - 3} = \frac{220 - 118}{2} = \frac{102}{2} = 51$

So:

$\frac{{x}^{3} - 5 x + 2}{{x}^{2} - 8 x + 15} = x + 8 - \frac{7}{x - 3} + \frac{51}{x - 5}$