This answer is directed at the comment that a proof that $R$ a UFD $\implies$ $R[t]$ a UFD takes two pages, i.e., one lecture two give.
In $\S 15.5$ of my commutative algebra notes I give a "lemmaless" proof of this fact modelled after the Lindemann-Zermelo direct proof that $\mathbb{Z}$ is a UFD. This proof shows up several times in the literature; the earliest I found was in a 1950 paper of S. Borofsky (but I wouldn't be surprised to hear that Hasse and/or Zermelo knew it, for instance). The proof takes a little over half a page, including displayed equations. It does use the decomposition of the UFD property into "ACCP" (ascending chain condition on principal ideals) plus "EL" (irreducible elements are prime), but the same can be said about the classic proof that $\mathbb{Z}$ is a UFD. All in all I think this would be a short amount of class time to cover some very important results, thus a good investment. Certainly I would rather do this than try to give some independent proof of the much weaker result that irreducible elements in polynomial rings over fields generate radical ideals.
