# What is an irreducible polynomial?

##### 1 Answer
Oct 24, 2015

An irreducible polynomial is one that cannot be factored into simpler (lower degree) polynomials using the kind of coefficients you are allowed to use, or is not factorisable at all.

#### Explanation:

Polynomials in a single variable

${x}^{2} - 2$ is irreducible over $\mathbb{Q}$. It has no simpler factors with rational coefficients.

${x}^{2} + 1$ is irreducible over $\mathbb{R}$. It has no simpler factors with Real coefficients.

The only polynomials in a single variable that are irreducible over $\mathbb{C}$ are linear ones.

Polynomials in more than one variable

If you are given a polynomial in two variables with all terms of the same degree, e.g. $a {x}^{2} + b x y + c {y}^{2}$, then you can factor it with the same coefficients you would use for $a {x}^{2} + b x + c$.

If it is not homogeneous then it may not be possible to factor it. For example, ${x}^{2} + x y + y + 1$ is irreducible.