Factoring

Factoring is the process of expressing an object as a product of its factors that multiply to give the original object. In mathematics, factoring refers to decomposing algebraic structures such as polynomials, integers, and matrices into simpler components, often revealing underlying structure and simplifying calculations. In abstract algebra, factoring is closely tied to the notion of unique factorization into irreducibles, up to units.

Polynomial factoring: The task of expressing a polynomial with coefficients in a given field or ring as a product of lower-degree polynomials. Over the rationals, polynomials factor into irreducibles with rational coefficients, and common techniques include identifying rational roots (Rational Root Theorem), factoring by grouping, and using special formulas such as difference of squares, perfect square trinomials, and sum/differences of cubes. Algorithms for more complex polynomials include Berlekamp–Zassenhaus and Cantor–Zassenhaus over finite fields.

Integer factoring: The integer factorization problem asks to write a composite integer as a product of primes. This problem is computationally hard in general and forms the security basis for RSA cryptography. Basic methods include trial division; advanced algorithms include Pollard's rho, Pollard's p−1, elliptic curve factorization, and the number field sieve.

Matrix factorization: In linear algebra, matrices may be factored into a product of simpler matrices, such as LU decomposition (lower and upper triangular factors), QR decomposition, Cholesky decomposition (for positive definite matrices), and singular value decomposition (SVD). These factorizations facilitate solving linear systems, eigenvalue computations, and data analysis.

Applications and notes: Factoring appears in solving equations, simplifying expressions, computer algebra, cryptography, and numerical methods. In many settings, factorization is unique up to units or ordering, reflecting the algebraic structure of the object.