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:
x2 + 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 (x + b)(x + c).
Example 2:
The polynomial
x2 – 2
is irreducible over the integers. However, we could factor it as
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!)
