Question

How do you write the partial fraction decomposition of the rational expression `(4x^2+3)/((x-5)^3)`?

Answer 1

`(4x^2+3)/(x-5)^3=4/(x-5)+40/(x-5)^2+103/(x-5)^3`

Explanation:

Let ,

`(4x^2+3)/(x-5)^3=A/(x-5)+B/(x-5)^2+C/(x-5)^3 ...tocolor(red)((M)`

`:.4x^2+3=A(x-5)^2+B(x-5)+C`

`:.4x^2+3=A(x^2-10x+25)+B(x-5)+C`

`:.4x^2+3=Ax^2-10Ax+25A+Bx-5B+C`

`:.4x^2+0*x+3=Ax^2+x(B-10A)+(25A-5B+C)`

Comparing coefficients of `x^2 ,x and` free term,we get

`color(blue)(A=4......................to(1)`

`B-10A=0.........,.to(2)`

`25A-5B+C=3..to(3)`

Substitute `color(blue)(A=4` into `(2)`

`:.B-10(4)=0=>color(blue)(B=40`

Again subst. `A=4 and B=40` into `(3)`

`:.25(4)-5(40)+C=3`

`:.C=3-100+200=>color(blue)(C=103`

Hence, from `color(red)((M))`,we have

`(4x^2+3)/(x-5)^3=color(blue)(4)/(x-5)+color(blue)(40)/(x-5)^2+color(blue)(103)/(x-5)^3`