factorable

Factorable is a term used in mathematics to describe an object that can be written as a product of two or more factors within a given algebraic structure. The most common contexts are integers and polynomials.

In integers, a positive integer is factorable if it has a nontrivial factorization into integers greater than 1, i.e., n = ab with 1 < a ≤ b < n. Numbers with such factorizations are composite; primes are not factorable. Factorability of integers relates to prime factorization and underpins many number-theoretic algorithms.

In polynomials, a polynomial f over a field F is factorable if it can be expressed as a product f(x) = g(x) h(x) with deg g, deg h ≥ 1 and coefficients in F. If no such factorization exists, f is irreducible over F. Factorability is field-dependent: a polynomial irreducible over Q may factor over R or C. For example, x^2 − 5x + 6 factors over Q as (x − 2)(x − 3), while x^2 + 1 is irreducible over R but factors over C as (x + i)(x − i). The concept extends to higher algebraic structures where elements factor into irreducibles within a domain, with unique factorization domains ensuring a unique factorization up to units and order.

Factorability also appears in computational contexts, including integer factorization and polynomial factorization algorithms, which are central to algebraic computation and number theory.

In summary, factorable describes the ability to decompose an object into a product of simpler factors, with the precise meaning depending on the chosen domain.