factorized

Factorized is an adjective describing an expression written as a product of factors. Factorization is the process of rewriting an object as a product of simpler or more fundamental constituents. In mathematics, factorization is used to simplify expressions, solve equations, and analyze structure.

In polynomials, factorization over a field means expressing a polynomial as a product of irreducible polynomials. The Fundamental Theorem of Algebra implies that over the complex numbers every nonconstant polynomial factors completely into linear factors, while over the rationals or integers polynomials may factor into irreducible components. Common methods include finding roots, factoring by grouping, and using special forms such as difference of squares, perfect square trinomials, or the sum and difference of cubes.

In number theory, integer factorization expresses a positive integer as a product of primes. The prime factorization is unique up to order, a fact known as the Fundamental Theorem of Arithmetic. Prime factorization is central to cryptography; many cryptographic systems rely on the difficulty of factoring large semiprimes.

In linear algebra, matrix factorization expresses a matrix as a product of matrices with particular properties, such as LU decomposition, QR factorization, Cholesky decomposition, or singular value decomposition (SVD). These factorizations facilitate solving linear systems, computing determinants or inverses, and analyzing matrix structure.

Other uses include factorization in computing, where factorized data can refer to codifying categorical information, or general algebraic practice of decomposing expressions into simpler factors. See also Factorization and related methods in polynomials, numbers, and matrices.