# Irreducible (Prime) Polynomials

A
polynomial
with integer coefficients that cannot be
factored
into polynomials of lower
degree
, also with integer coefficients, is called an
**
irreducible
**
or
**
prime polynomial
**
.

**
Example 1:
**

${x}^{2}+x+1$

is an irreducible polynomial. There is no way to find two integers $b$ and $c$ such that their product is $1$ and their sum is also $1$ , so we cannot factor into linear terms $\left(x+b\right)\left(x+c\right)$ .

**
Example 2:
**

The polynomial

${x}^{2}-2$

is irreducible over the integers. However, we could factor it as

$\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)$

if we are allowed to use irrational numbers. So the irreducibility of a polynomial depends on the number system you're working in.

(When you study
complex numbers
, you'll find that the only irreducible polynomials over
**
C
**
are the degree
$1$
polynomials!)