Factorization

Factorization is the process of expressing an object as a product of smaller or simpler factors. In mathematics, factorization often refers to breaking numbers into prime factors and factoring polynomials into irreducible factors, but the concept extends to other algebraic structures as well. It is a way of revealing the building blocks that combine to form the original object.

In number theory, prime factorization expresses a positive integer as a product of prime numbers raised to powers. By the fundamental theorem of arithmetic, every positive integer has a unique prime factorization (up to ordering). For example, 360 = 2^3 × 3^2 × 5. Factorization underpins many algorithms and problems, including gcds, lcms, and modular arithmetic. Some integers are easy to factor, while others are computationally difficult, a fact exploited in public-key cryptography.

In algebra, polynomial factorization writes a polynomial as a product of polynomials of lower degree, ideally irreducible factors. Over a field, every polynomial can be factored into irreducibles, similar to prime numbers in integers. Techniques include extracting common factors, factoring by grouping, and applying formulas for special products such as difference of squares, perfect square trinomials, and sum or difference of cubes. Over the rationals or integers, Gauss’s lemma helps relate factorizations in different coefficient domains. Over the real or complex numbers, polynomials factor further into linear or quadratic factors, respectively.

Factorization also appears in linear algebra and other areas, where objects are decomposed into simpler components. The concept is central to solving equations, simplifying expressions, and understanding the structure of mathematical objects.