A polynomial consists of a sum of monomials. However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros. The process of rewriting a polynomial as a product is called factoring.
Factoring and Expanding Polynomials
Factoring is the decomposition of an algebraic object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original. As an example, the integer
For example:
is a factorization of a polynomial of degree
The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials. (These are polynomials which cannot be factored non-trivially.)
The inverse procedure of polynomial factorization is expansion, which is just explicitly writing out the multiplication of two or more factors, for example:
Example: Factoring by Grouping
One way to factor polynomials is factoring by grouping. This is done by grouping the terms in the polynomial into two or more groups in such a way that each group can be factored separately. The results of these factorizations can sometimes be combined to make an even more simplified expression. For example, to factor the polynomial
As both terms in the left expression are divisible by
Both groups share the same factor
Sometimes, when factoring a polynomial in two or more variables, this last step is not possible and we have to content ourselves with having two or more terms which are each factorized themselves: