A non-constant polynomial, $P(x),$ is said to be an irreducible polynomial or prime polynomial over a set of numbers, if it cannot be factored into the product of two non-constant polynomials over the same set of numbers.

When a polynomial is not prime, it is called a reducible polynomial. Keep in mind that a given polynomial can be prime over one set of numbers but reducible over another. Over the set of the complex numbers, only the linear polynomials are prime. This can be proven by using the Fundamental Theorem of Algebra.