Question

Write the partial fraction decomposition of the rational expression 3x-5/x²-4x-32

Answer 1

`(3x-5)/(x^2-4x-32)=(3x-5)/((x+4)(x-8))`

Denominator has been factorized on the right.

Now we need to write this fraction as a sum of two fraction.

`(3x-5)/((x+4)(x-8))=(A)/((x+4))+(B)/((x-8))=(A(x+4)+B(x-8))/((x+4)(x-8))`
`A(x+4)+B(x-8)=3x-5\text{ at this stage we will find A B constants}`

when x = 8 we have following

`A(8+4)+B(8-8)=3\cdot8-5\rightarrow12A+0=19\rightarrow A=(19)/(12)\approx1.58`

Similarly when x =-4 A = 17/12 these are the constants