Problem:
If we have a polynomial $f$ with a derivative f\,' and quotient $q$ function defined as:
q(x)=\sum_{i=1}^{\infty}a_ix^{-i}=\frac{f\,'(x)}{f(x)},
and the roots of $f$ are $x_1,x_2,\ldots,x_k$, how to prove
$a_i=\sum_{j=1}^{k}x_j^i$
Details:
If $f(x)=x^2-5x+6$, f\,'(x)=2x+5,
$q(x)=2 x^{-1}+5 x^{-2}+13 x^{-3}+35 x^{-4}+97 x^{-5}+\ldots$