So I was faced by this problem: If $p,q \in K[x]$, with $q = q_{0}^{\alpha_{0}} \dots q_{m}^{\alpha_{m}}$, all of $q_{i}$ irreducible and $\alpha_{i}$ natural numbers. So, there is some $p_{0} \dots p_{m} \in K[x]$ where $$ \frac{p}{q}=\sum_{j = 0}^{m}\frac{p_{j}}{q_{j}^{\alpha_{j}}}. $$
So, at first sight, it looks a lot like the partial fractions decomposition, which we use a lot in Calculus, but I can't prove this. Any help, I will be appreciated. Thanks in advance!
