Question

How do you express `(x^5 + 1)/(x^6 - x^4)` in partial fractions?

Answer 1

`(x^5+1)/(x^4(x+1)(x-1)) = -1/x^2 -1/x^4+1/(x-1)`

Explanation:

`(x^5+1)/(x^4(x^2-1)) =(x^5+1)/(x^4(x+1)(x-1))`
`(x^5+1)/(x^4(x+1)(x-1))= A/x +B/x^2+C/x^3+D/x^4 +E/(x+1)+F/(x-1)`

`x^5+1=A(x^3)(x^2-1)+B(x^2)(x^2-1)+C(x)(x^2-1)+D(x^2-1)+E(x^4)(x-1)+F(x^4)(x+1)`

`x^5+1=Ax^5-Ax^3+Bx^4-Bx^2+Cx^3-Cx+Dx^2-D+Ex^5-Ex^4+Fx^5+Fx^4`

`1=A+E+F, 0=B-E+F, 0=-A+C,0=-B+D, 0=-C, 1=-D`

`A=0,B=-1,C=0,D=-1,E=0,F=1`

`(x^5+1)/(x^4(x+1)(x-1)) = -1/x^2 -1/x^4+1/(x-1)`