factored

Factored is the past participle of factor, used to describe an object that has been expressed as a product of factors. In mathematics, to factor a number or an expression means to rewrite it as a product of simpler components.

For integers, factoring means expressing a number as a product of primes. The prime factorization of 60 is 2 × 2 × 3 × 5. The fundamental theorem of arithmetic states that every integer greater than 1 has a unique prime factorization up to order.

For polynomials, factoring expresses the polynomial as a product of polynomials of smaller degree, such as x^2 − 5x + 6 factoring to (x − 2)(x − 3). Factored form can help solve equations, find roots, and simplify expressions. Over a given domain (such as the integers or the reals), a polynomial may factor into irreducible factors that cannot be further factored in that domain.

In abstract algebra, factorization extends to rings and modules, where elements are factored into irreducibles or primes. In many rings, including the integers, a fundamental property is the existence and uniqueness (up to units and order) of such factorizations.

Factored representations have practical applications, including solving algebraic equations, simplifying rational expressions, and in computer algorithms for symbolic computation; in number theory, prime factorization underpins cryptographic systems such as RSA, where the difficulty of factoring large numbers provides security.

See also: factor, factorization, prime factorization, irreducible polynomial, RSA.

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