Suppose I have a potential as form $$ V(r)=\frac{\sum_{k=0}^{K}a_k r^k}{\sum_{m=0}^{M}b_m r^m} \tag{1} $$ How can I match the given potential $$ V(r)=\frac{-Z}{r}+\alpha r+\beta r^2 \tag{2} $$ with equation (1)? I mean how can I find the values of K and M for equation (2)?
(They were obtained 3 and 1 for K and M respectively in a reference but I don't understand HOW?!!)
Hint: Consider \begin{align*} V(r)=\frac{-Z+\alpha r^2+\beta r^3}{r} \end{align*}