# Find the values of a and b that make the following expression an identity? (5x+31)/((x−5)(x+2) )= (a)/(x−5) − (b)/(x+2)

Apr 6, 2017

$a = 8$. $b = 3$ and identity is

$\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{8}{x - 5} - \frac{3}{x + 2}$

#### Explanation:

This is a typical example of Partial-Fraction Decomposition

As $\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{a}{x - 5} - \frac{b}{x + 2}$,

we can write RHS as

$\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{a \left(x + 2\right) - b \left(x - 5\right)}{\left(x + 2\right) \left(x - 5\right)}$

or $\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{a x + 2 a - b x + 5 b}{\left(x + 2\right) \left(x - 5\right)}$

or $\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{\left(a - b\right) x + \left(2 a + 5 b\right)}{\left(x + 2\right) \left(x - 5\right)}$

i.e. $a - b = 5$ and $2 a + 5 b = 31$

Solving these simultaneous equations, by multiplying first by $5$ and adding to second equation, we get

$7 a = 56$ or $a = 8$ and then $b = 3$ and identity is $\frac{5 x + 31}{\left(x - 5\right) \left(x + 2\right)} = \frac{8}{x - 5} - \frac{3}{x + 2}$