Every prime ideal in $\mathbb{Z}[x]$ is generated by at most two elements

Question

I am trying to prove

Every prime ideal in $\mathbb{Z}[x]$ can be generated by at most two elements.

Since I have not seen any prime ideal generated by 3 elements, this statement seems true to me. But I cannot find a way to prove it. Thanks for any help in advance!

Answer 1

(This is more "things that work in $\mathbb{Z}[x]$" more than "general algebraic methods", but it works.)

Suppose you have three (or more) generators. Then either two of them are constants or two of them are polynomials (of positive degree). With what can you replace these two generators? (Hint: Same answer for both cases.)