[Math] Simply put, what are the similarities between integers and polynomials

Question

The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of the Euclidean domain. It isn't obvious to me how this is the case. Could you provide a palatable explanation?

Answer 1

absolute value of integer <-> degree of polynomial

positive integer <-> monic polynomial

+/- 1 <-> constant polynomial

prime integer <-> irreducible polynomial

With these correspondences, there are many identical notions and theorems, like the division algorithm, unique prime factorization, principal ideals, LCM, GCD, ...