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)

1 Answer
Apr 6, 2017

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)