# How do you express  (3x^2 - 4x - 2) / [(x-1)(x-2)] in partial fractions?

Aug 7, 2016

$\frac{3 {x}^{2} - 4 x - 2}{\left(x - 1\right) \left(x - 2\right)} = 3 + \frac{3}{x - 1} + \frac{2}{x - 2}$

#### Explanation:

$\frac{3 {x}^{2} - 4 x - 2}{\left(x - 1\right) \left(x - 2\right)}$

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

$= \frac{3 {x}^{2} - 9 x + 6 + 5 x - 8}{{x}^{2} - 3 x + 2}$

$= \frac{3 \left({x}^{2} - 3 x + 2\right) + 5 x - 8}{{x}^{2} - 3 x + 2}$

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

$= 3 + \frac{5 x - 8}{\left(x - 1\right) \left(x - 2\right)}$

$= 3 + \frac{A}{x - 1} + \frac{B}{x - 2}$

Use Heaviside's cover-up method to find:

$A = \frac{5 \left(1\right) - 8}{\left(1\right) - 2} = \frac{- 3}{- 1} = 3$

$B = \frac{5 \left(2\right) - 8}{\left(2\right) - 1} = \frac{2}{1} = 2$

So:

$\frac{3 {x}^{2} - 4 x - 2}{\left(x - 1\right) \left(x - 2\right)} = 3 + \frac{3}{x - 1} + \frac{2}{x - 2}$