What Are Irreducible Polynomials?

Irreducible polynomials

A polynomial is irreducible if it cannot be factored into two or more non-constant polynomials of a given type.

A polynomial's irreducibility depends on what kinds of polynomial factors you would allow; for example, $2x^2+12x+18$ is not irreducible if the factors need integer coefficients. $$2x^2+12x+18\rightarrow (2x+4)(x+3)$$ But $4x^2+4x-3$ is irreducible with this condition.

However, if you were factoring with rational coefficients, then $4x^2+4x-3$ is not irreducible: $$4x^2+4x-3\rightarrow (4x+6)(x-\frac{1}{2})$$

Some conventions

As to the field (type of number) the coefficients belong in, one would say, "The polynomial $P$ is irreducible over this field".

For example, $4x^2+4x-3$ is irreducible over the integers.