Let $R$ be a unique factorization domain which is a finitely generated $\Bbbk$-algebra for an algebraically closed field $\Bbbk$. For $x\in R\setminus\{0\}$, let $y$ be an $n$-th root of $x$. My question is, is the ring
$ A := R[y] := R[T]/(T^n - x) $
a unique factorization domain as well?
Edit: I know the classic counterexample $\mathbb{Z}[\sqrt{5}]$, but $\mathbb{Z}$ it does not contain an algebraically closed field. I am wondering if that changes anything.
Edit: As Gerry's Answer shows, this is not true in general. What if $x$ is prime? What if it is a unit?