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)#

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Answer 1 · 2017-04-06T06:59:24

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

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

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)#