Question

How do you express `(x^3+x^2+2x+1)/((x^2+1)(x^2+2))` in partial fractions?

Answer 1

`(x^3+x^2+2x+1)/((x^2+1)(x^2+2))=x/(x^2+1) + 1/(x^2+2)`

Explanation:

Neither of the quadratic factors in the denominator have linear factors with Real coefficients.

So we are looking for a partial fraction decomposition of the form:

`(x^3+x^2+2x+1)/((x^2+1)(x^2+2))`

`=(Ax+B)/(x^2+1) + (Cx+D)/(x^2+2)`

`=((Ax+B)(x^2+2) + (Cx+D)(x^2+1))/((x^2+1)(x^2+2))`

`=((A+C)x^3+(B+D)x^2+(2A+C)x+(2B+D))/((x^2+1)(x^2+2))`

Equating coefficients we get the following system of linear equations:

`{ (A+C=1), (B+D=1), (2A+C=2), (2B+D=1) :}`

From the first and third equations we find `A=1`, `C=0`

From the second and fourth equations we find `B=0`, `D=1`

So:

`(x^3+x^2+2x+1)/((x^2+1)(x^2+2))=x/(x^2+1) + 1/(x^2+2)`