Section 3.4
Factorization
The role of prime numbers in the case of integers is played by irreducible polynomials in the context of polynomial arithmetic.
In the following R is, without explicit mention of the contrary, always a field, like Q, R, C or Z/pZ with p prime. These arithmetic systems have in common that every nonzero element has a multiplicative inverse.
First, zeros of a polynomial are related to linear factors (that is, factors of degree 1).
Let f 
R[X].
An element x R is a zero of
f if and only if X - x divides f.
Here is the counterpart in the setting of polynomial rings of primality.
A polynomial f
R[X] is called irreducible if deg(f) > 0 and
if the only nonconstant polynomials g with g | f have
degree deg(f); in other words, if the only divisors
of f are the constants and the constant multiples of f.
If f is not irreducible, f is called reducible.
We shall study factorizations of a polynomial, that is, ways to write the
polynomial as a product of polynomials of smaller degree.