Factorisation

Factorisation is the process of breaking down a mathematical expression into a product of other expressions. It is a fundamental technique in algebra and number theory, with applications in various fields such as computer science, cryptography, and engineering. The goal of factorisation is to express a given expression as a product of simpler factors, which can often reveal important properties and relationships.

In algebra, factorisation is commonly used to simplify expressions, solve equations, and analyze functions. For example, the expression x^2 - 4 can be factored into (x + 2)(x - 2), which reveals that the expression has roots at x = 2 and x = -2. This technique is also used in polynomial factorisation, where a polynomial is expressed as a product of simpler polynomials.

In number theory, factorisation is used to determine the prime factors of a given number. The Fundamental Theorem of Arithmetic states that every integer greater than 1 can be uniquely expressed as a product of prime numbers. For example, the number 60 can be factored into 2^2 * 3 * 5, which reveals its prime factors.

Factorisation can also be applied to matrices and other mathematical structures. In linear algebra, factorisation techniques such as LU decomposition and QR decomposition are used to solve systems of linear equations and analyze the properties of matrices.

Overall, factorisation is a powerful and versatile technique that plays a crucial role in many areas of mathematics and its applications. By breaking down complex expressions into simpler factors, factorisation can reveal important properties, simplify calculations, and provide insights into the underlying structure of mathematical objects.