The determination of a set of factors (divisors) of a given integer ("prime factorization"), polynomial ("polynomial factorization"), etc., which, when multiplied together, give the original number, polynomial, etc In many cases of interest (particularly prime factorization, factorization is unique, and so gives the "simplest" representation of a given quantity in terms of smaller parts.
The terms "factorization" and "factoring" are used synonymously.
The term "factorization" is occasionally misused, including by no less "authority" than The New York Times, where Fox (2006) wrote, "He was 88, which can be factored as 1, 2, 4, 8, 11, 22, 44, and 88." This usage is incorrect since the given numbers are indeed factors, but the collection of factors does not comprise a factorization.