factorizing

Factorizing, or factorization, is the process of rewriting a mathematical object as a product of its factors. In number theory, factorization typically refers to expressing an integer as a product of primes. In algebra, it means factoring a polynomial or algebraic expression into simpler polynomials or monomials. Factorization is the reverse operation of multiplication and is used to simplify expressions, solve equations, and study the structure of numbers and functions.

For integers, the Fundamental Theorem of Arithmetic states that every positive integer has a unique prime factorization, up to order and sign. Prime factorization can be obtained by trial division, or by more advanced algorithms for large numbers. For example, 360 = 2^3 · 3^2 · 5.

For polynomials, factorization seeks a product of polynomials with smaller degree that equals the original. Over the integers or rationals, common methods include factoring out a common factor, factoring by grouping, and recognizing patterns such as the difference of squares, perfect square trinomials, and sum or difference of cubes. Quadratic polynomials can often be factored by finding numbers that multiply to ac and sum to b. The Rational Root Theorem helps locate potential linear factors, after which synthetic division or polynomial long division can be used to extract factors. More complex polynomials may require substitution, advanced formulas, or irreducibility tests; in many domains, factorization is unique up to multiplication by units and ordering.

Applications include solving equations, expanding and simplifying expressions, and in cryptography and computer algebra. While easy for small cases, factorization becomes computationally challenging for large integers, which underpins some security systems.