What does factorization mean?
Factorization is the process of expressing a number or mathematical expression as a product of simpler components called factors. In arithmetic, it most often means finding the prime factorization — rewriting a number such as 60 as 2 × 2 × 3 × 5. In algebra, it involves rewriting polynomials like x² − 9 as products such as (x + 3)(x − 3), a technique essential for solving equations and simplifying expressions. In advanced mathematics and computing, the term extends to matrix factorizations, where a matrix is expressed as a product of matrices with special properties, underpinning numerical methods and machine learning. Factorization is central to number theory: the fact that every integer greater than 1 has a unique prime factorization is known as the Fundamental Theorem of Arithmetic. The word carries no emotional connotation; it is neutral and technical. Its practical importance is striking — modern cryptographic systems rely on the fact that multiplying two large primes is easy, while reversing the multiplication through factorization is computationally infeasible.
nounThe process of breaking down a number, polynomial, or other mathematical object into a product of simpler components (factors), or the expression that results from this process.
- The decomposition of a number into a product of its factors, especially prime factors.
- The decomposition of a polynomial or algebraic expression into a product of simpler expressions.
- In linear algebra and computing, the representation of a matrix as a product of matrices with special properties.
"Prime factorization of 60 gives 2 × 2 × 3 × 5."
"Students learn factorization early in algebra, when they first rewrite x² − 9 as (x + 3)(x − 3)."
"The security of RSA encryption depends on the difficulty of the factorization of very large numbers."
Used when referring to multiple distinct decompositions, e.g. different factorizations of the same polynomial over different number systems.
"There are several possible LU factorizations depending on the pivoting strategy."
Every whole number greater than 1 has exactly one prime factorization — a uniqueness so fundamental it's called the Fundamental Theorem of Arithmetic.
Reviewed by Deb Chak, Editor. AI-assisted content curated by RJS Tech Solutions LLP.
Etymology of factorization
Factorization derives from 'factor', which comes from the Latin 'factor', meaning 'one who does or makes', from the verb 'facere' ('to do' or 'to make'). The mathematical sense of factor as a component in a product emerged from the idea of something that 'makes up' a larger whole. The suffix '-ization' marks the noun of action, so factorization literally denotes the act of resolving something into its factors. Cognates sharing the Latin root include 'factory', 'faction', and 'factitious'.
Related word forms
How factorization is actually used
A formal, primarily mathematical term used in education, research mathematics, computer science, and cryptography. In American usage the verb is usually 'factor' or 'factorize'; in British usage 'factorise' and 'factorisation' with -ise/-isation spellings are standard.
Easily confused with factorization
Fractionalization means dividing something into fractions or factions, whereas factorization is specifically the mathematical decomposition of an expression into factors.
Fractalization refers to creating fractal patterns, while factorization refers to expressing a mathematical object as a product of factors.