Unique Factoring in $\mathbb{Z}[x]$

Question

While in class, my professor gave us a proposition that if $R$ is a unique factoring domain, then $R[x]$ is also a unique factoring domain. While in class I didn't think too much about it, but now I think I don't understand what he meant because of the following example: $$x^6=x^3 \cdot x^3$$ $$x^6=x^4 \cdot x^2$$ Clearly, these factorizations aren't unquie, but $\mathbb{Z}$ is a unique factoring domain, so shouldn't $\mathbb{Z}[x]$ be? Is it the case I simply wrote the theorem down incorrectly, or am I misunderstanding what it means to be a unique factorization domain?