Question

How do you express `(2x^4 + 9x^2 + x - 4)/(x^3 + 4x)` in partial fractions?

Answer 1

`(2x^4+9x^2+x-4)/(x^3+4x)=2x-1/x+(2x+1)/(x^2+4)`

Explanation:

Let us first divide `2x^4+9x^2+x-4` by `x^3+4x` to get the degree of numerator less than that of denominator and we get

`(2x^4+9x^2+x-4)/(x^3+4x)=2x+(x^2+x-4)/(x(x^2+4))`

Let us now get partial fractions of `(x^2+x-4)/(x(x^2+4))`, which will be of the form

`(x^2+x-4)/(x(x^2+4))=A/x+(Bx+C)/(x^2+4)`

or `x^2+x-4=A(x^2+4)+x(Bx+C)=(A+B)x^2+Cx+4A`

andcomparing coefficients of like powers

`4A=-4` or `A=-1`, `C=1` and

`A+B=1` i.e. `B=1-A=2`

Hence `(2x^4+9x^2+x-4)/(x^3+4x)=2x-1/x+(2x+1)/(x^2+4)`