Well it turns out the link I gave is not directly applicable since $\displaystyle \,\mathbb{Z}$ is not a field. But $\displaystyle \,\mathbb{Z}$ is a unique factorization domain (UFD), and it can be proven that the ring of polynomials over a UFD is also a UFD. Apologies for not noticing this.

I am looking into the best way to present a proof and/or give a reference. It should be noted that I am not an expert in this but am also learning, as time permits. I try to make sure everything I post is well-reasoned and accurate, but of course, it doesn't always work out quite like that.

I hope to post again before too many hours pass with something useful; I am busy at the moment.