factorization
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Definition
Factorization (noun) 1. (Mathematics) The process or result of resolving a mathematical entity (such as a number or a polynomial) into a product of other, simpler entities (called factors), such that when these factors are multiplied together, they yield the original entity.
Usage
Factorization is a fundamental concept in arithmetic and algebra. It is used to simplify expressions, solve equations, and understand the properties of numbers and polynomials. - The factorization of the number 12 into prime factors is 2 × 2 × 3. - Finding the factorization of a quadratic polynomial is a common step in solving quadratic equations.
Examples
Advanced Usage
- Unique Factorization Domain (UFD): An advanced algebraic structure in which an element's factorization into irreducible elements is unique (up to order and units).
- Matrix Factorization: In linear algebra, decomposing a matrix into a product of matrices, such as in LU factorization or QR factorization.
- Integer Factorization: The decomposition of a composite number into a product of smaller integers, a problem of great importance in cryptography.
Variants and Related Words
- Factor (verb/noun): To break down into factors; a number or expression that divides another without a remainder.
- Factorize (verb): To perform factorization. (e.g., )
- Factoring (noun): Often used synonymously with factorization, especially in an educational context. (e.g., )
- Prime Factorization: The process of expressing a number as a product of its prime factors.
Synonyms
- Decomposition (into factors)
- Resolving into factors
- Factoring
Related Phrases
- Complete Factorization: The process of factoring an expression until all remaining factors are prime (for numbers) or irreducible (for polynomials).
- Factorization Theorem: A theorem stating conditions under which factorization is possible or unique.
Noun
- (mathematics) the resolution of an entity into factors such that when multiplied together they give the original entity