# Factorization

In mathematics, **factorization** or **factoring** is the decomposition of an object into a list of (smaller) 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 both 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.

See also: Prime factorization algorithm, program synthesis

## External links

- A page about factorisation, Algebra, Factoring
- WIMS Factoris is an online factorization tool.