What is an irreducible polynomial?

1 Answer
Oct 24, 2015

Answer:

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 #QQ#. It has no simpler factors with rational coefficients.

#x^2+1# is irreducible over #RR#. It has no simpler factors with Real coefficients.

The only polynomials in a single variable that are irreducible over #CC# 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. #ax^2+bxy+cy^2#, then you can factor it with the same coefficients you would use for #ax^2+bx+c#.

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