factorisations

Factorisation, or factorization, is the process of expressing an object as a product of its factors. In mathematics, this most often means breaking numbers or polynomials into products of smaller pieces that multiply to the original value, but the idea applies to other structures such as matrices and abstract algebras as well.

Integer factorisation concerns expressing a positive integer as a product of prime numbers. The fundamental theorem of arithmetic states that every positive integer has a unique prime factorisation up to the order of the factors. For example, 360 = 2^3 × 3^2 × 5. Factorisation can also include a sign, so negative numbers factor as -1 times a positive integer’s prime factorisation.

Polynomial factorisation is the expression of a polynomial as a product of polynomials of lower degree with coefficients in a given domain (such as the integers or rationals). Over the integers, factoring uses methods such as finding rational roots, grouping, and applying special formulas (difference of squares, perfect square trinomials, sum or difference of cubes). A polynomial is irreducible over a domain if it cannot be factored nontrivially there. Example: x^2 − 5x + 6 factors as (x − 2)(x − 3) over the integers.

Matrix factorisation decomposes a matrix into a product of simpler matrices, revealing structure and enabling computations. Common types include LU decomposition (lower times upper), QR decomposition (orthogonal times upper), and the singular value decomposition (UΣV^T).

In abstract algebra, factors relate to prime elements and irreducibles. In a unique factorisation domain, every element factors uniquely into irreducibles up to units and order.

Applications include solving equations, simplifying expressions, and cryptography, where the difficulty of integer factorisation underpins some security systems.