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

Feb 24, 2016

$\frac{2 x - 2}{\left(x - 5\right) \left(x - 3\right)} \text{ "=" } \frac{4}{x - 5} - \frac{2}{x - 3}$

#### Explanation:

Write as:$\text{ " (2x-2)/((x-5)(x-3))" "=" } \frac{A}{x - 5} + \frac{B}{x - 3}$

Thus: $\text{ } \frac{2 x - 2}{\left(x - 5\right) \left(x - 3\right)} = \frac{A \left(x - 3\right) + B \left(x - 5\right)}{\left(x - 5\right) \left(x - 3\right)}$

So:$\text{ "2x-2" "=" } A \left(x - 3\right) + B \left(x - 5\right)$

$2 x - 2 \text{ "=" } A x - 3 A + B x - 5 B$

Collecting like terms

$2 x - 2 \text{ "=" } \left(A + B\right) x - 3 A - 5 B$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Comparing LHS to RHS

$2 x = \left(A + B\right) x \text{ so "A+B =2" }$............................(1)

$- 2 = - 3 A - 5 B \text{ so "B=(2-3A)/5" }$...................(2)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (2) into (1) giving:

$A + \frac{2 - 3 A}{5} = 2$

$\frac{5 A + 2 - 3 A}{5} = 2$

$2 A = \left(2 \times 5\right) - 2 = 8$

$A = 4 \text{ }$.......................................(3)

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Substitute (3) into (1) giving:

$4 + B = 2$

$B = - 2$
'~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{b l u e}{\frac{2 x - 2}{\left(x - 5\right) \left(x - 3\right)} \text{ "=" } \frac{4}{x - 5} - \frac{2}{x - 3}}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Check: $4 \left(x - 3\right) - 2 \left(x - 5\right) = 4 x - 12 - 2 x + 10 = 2 x - 2$

Matching original numerator so ok!