Calculate partial fractions

Question

calculate partial fractions for: $1/x^2(x^2 + 1)$ I have tried solving by expanding it like this: $A/x^2 + B/ (x^2 + 1)$ and it results in the right answer as given in class. But partial fractions expansion rules suggest that I have to expand it like this: $A/x + B/x^2 + (Cx + D)/(x^2 + 1)$ right? Then how do I solve through the above expansion?

Answer 1

Clear fractions to get $Ax(x^2+1)+B(x^2+1)+(Cx+D)x^2=1$

Now, $x=0$ immediately yields $B=1$

Next let $x=i$. This gives you $Ci+D=-1$ from which you get $C=0$ and $D=-1$.

Finally, let $x=1$ to get $A=0$.

Answer 2

Just to add a small point to mrtwhs' answer, I'd note that the second expansion you gave should be used here and that it just happened to be the case in this circumstance that the terms you omitted in the first version had zero coefficient. That is, the first expression is missing the $\frac{1}{x}$ and $\frac{x}{x^2+1}$ terms. This ends up being ok in this case because, as mrtwhs' showed, the coefficients of those terms end up being zero. If the numerator weren't a constant and you had something like $\frac{x^2+3x+1}{x^2\left(x^2+1\right)}$ you'd end up with an incorrect expansion if those terms were again omitted.