# Factorization

*This article is about the mathematical concept. For the financial term see Factoring (trade).*

In mathematics, **factorization** or **factoring** is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or **factors**, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5; and the polynomial *x*^{2} − 4 factors as (*x* − 2)(*x* + 2). In all cases, we obtain a product of simpler things.

The aim of factoring is usually to reduce something to "basic building blocks", such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra.

Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.

A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.

Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function.

## Factoring in mathematical logic

In mathematical logic and automated theorem proving, factoring is the technique of deriving a single, more specific atom from a disjunction of two more general unifiable atoms. For example, from ∀ *X*, *Y* : *P*(*X*, *a*) or *P*(*b*, *Y*) we can derive *P*(*b*, *a*).

## See also

## External links

- A page about factorization, Algebra, Factoring
- WIMS Factoris is an online factorization tool.
- Polynomial Factoring is a comprehensive tutorial resource on basic factoring of polynomials.

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