If $a+bc$ and $c$ have irreducible factors in common, then $a$ and $c$ have the same irreducible factors in common.

Question

Let $a,b,c \in K[t]$ where $K$ is field with characteristic not $2$ or $3$ and see title for the question. This is a problem I encountered in showing the image of some map is finite, which I need in my thesis. I tried decomposing in irreducible elements, but I got stuck. Any help would be appreciated.

Answer 1

No. Take $a=6$, and $c=2$. Then $a+bc$ and $c$ have the factor $2$ in common, but $a$ has the factor $3$ that's not common to $c$.

These are integers, you'd object. But it's the same over any UFD with at least two distinct irreducible elements $p$ and $q$. Take $a=pq$ and $c=p$.

The ring $K[t]$ has infinitely many irreducible elements, so you're doomed.