Question

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)`

Answer 1

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

`(5x+31)/((x-5)(x+2))=8/(x-5)-3/(x+2)`

Explanation:

This is a typical example of Partial-Fraction Decomposition

As `(5x+31)/((x-5)(x+2))=a/(x-5)-b/(x+2)`,

we can write RHS as

`(5x+31)/((x-5)(x+2))=(a(x+2)-b(x-5))/((x+2)(x-5))`

or `(5x+31)/((x-5)(x+2))=(ax+2a-bx+5b)/((x+2)(x-5))`

or `(5x+31)/((x-5)(x+2))=((a-b)x+(2a+5b))/((x+2)(x-5))`

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

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

`7a=56` or `a=8` and then `b=3` and identity is `(5x+31)/((x-5)(x+2))=8/(x-5)-3/(x+2)`