Let R be a field, like Q, R, C or Z/pZ with p prime, and let a, b R[X].

Definition


The polynomial b is called a divisor of a if there exists a polynomial q R[X] such that a = qb. We use the notation b | a. If b 0, the polynomial q is unique and is called the quotient of a and b, and is denoted by a/b.

Instead of b is a divisor of a, we also say

a is a multiple of b, or a is divisible by b, or b is a factor of a, or b divides a.


Just like for the integers there is also a division algorithm for polynomials. It can be used to determine the greatest common divisor of two polynomials. The details are as follows.

Theorem

Let a and b be two polynomials in R[X] with b 0. There are polynomials q (the quotient) and r (the remainder) such that

a = qb + r   and   deg(r) < deg(b).

The polynomials q and r are uniquely determined.


Notation:
The remainder r is, just like for integers, denoted by a mod b.