A simple pole is not regular

Let $K$ be a field and $p,q\in K[X]$ with $p$ irreducible and $p\nmid q$. Let $c\in K$ be nonzero. Then the rational function $c\,q/p$ has a genuine pole at $p$: there are no polynomials $a,b$ with $p\nmid b$ and $c\,\dfrac{q}{p}\,b=a$ in $K(X)$.

This says that a nonzero multiple of a function $q/p$ with a simple pole cannot be rewritten with a denominator coprime to $p$.

Formalization Note The statement is purely algebraic; $c$ is viewed in $K(X)$ via the canonical embedding.