Question

How do you express `(x^4+1)/(x^5+6 x^3)` in partial fractions?

Answer 1

`(x^4+1)/(x^5+6x^3) = -1/(36x)+1/(6x^3)+(37x)/(36(x^2+6))`

Explanation:

`(x^4+1)/(x^5+6x^3) = (x^4+1)/(x^3(x^2+6))`

`color(white)((x^4+1)/(x^5+6x^3)) = A/x+B/x^2+C/x^3+(Dx+E)/(x^2+6)`

`color(white)((x^4+1)/(x^5+6x^3)) = (Ax^2(x^2+6)+Bx(x^2+6)+C(x^2+6)+Dx^4+Ex^3)/(x^2+6)`

`color(white)((x^4+1)/(x^5+6x^3)) = ((A+D)x^4+(B+E)x^3+(6A+C)x^2+6Bx+6C)/(x^2+6)`

Equating coefficients gives us this system of linear equations:

`{ (A+D = 1), (B+E = 0), (6A+C = 0), (6B = 0), (6C = 1) :}`

Hence:

`{ (A = -1/36), (B = 0), (C = 1/6), (D = 37/36), (E = 0) :}`

So:

`(x^4+1)/(x^5+6x^3) = -1/(36x)+1/(6x^3)+(37x)/(36(x^2+6))`