factorisé

Factorisé is the French term describing an expression that has been rewritten as a product of factors. In algebra, the act of factorising a polynomial or an integer yields a product form that is often easier to analyze, simplify, or solve. The resulting expression is called a factored form, and the factors reveal useful information such as common factors, roots, or divisibility properties.

In polynomial factorisation, the objective is to express a given polynomial as a product of polynomials of lower degree, ideally irreducible with respect to a chosen domain (for example, integers, rationals, reals, or complex numbers). The zero-product property plays a central role: if a product equals zero, then at least one factor must be zero, which helps to find roots of the polynomial. Common techniques include extracting a common factor, factoring by grouping, factoring trinomials, and special identities such as the difference of squares, perfect square trinomials, and the sum or difference of cubes.

Factorisation also applies to integers, where a number is written as its prime factorisation into prime factors raised to powers. This form is fundamental in number theory and has applications in solving Diophantine problems and in algorithms.

The possibility and form of factorisation depend on the chosen domain. Over the complex numbers, every non-constant polynomial factors completely into linear factors; over the rationals or integers, a polynomial may factor into irreducible polynomials that cannot be further decomposed within that domain.

In usage, factorisé is commonly encountered in mathematics education, symbolic computation, and problem solving as a tool to simplify expressions and identify roots or divisors.