Polynomiale

Polynomiale is a mathematical object also known as a polynomial. It is an algebraic expression built from constants and variables using only addition, subtraction, multiplication, and non-negative integer exponents. In its standard form a univariate polynomial in a variable x with coefficients from a ring R is written as P(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0, where a_n ≠ 0 and n is the degree of the polynomial. Polynomials can also involve several variables, for example P(x, y) = 3x^2y + 5xy^2 + 7.

Polynomials are closed under the operations of addition and multiplication, and they form a algebraic structure known as a polynomial ring. In one variable over a field F, the set F[x] is a principal ideal domain and a unique factorization domain, with important consequences for division and factorization. Key results include the division algorithm, which guarantees a quotient and a remainder when dividing by a nonzero polynomial, and the factor theorem, which connects roots to factors. Over the complex numbers, the Fundamental Theorem of Algebra asserts that every non-constant polynomial of degree n has exactly n roots in the complex plane counting multiplicities.

Polynomials are widely used in mathematics and applied sciences. They are employed in solving equations, interpolation (such as constructing polynomials that pass through a given set of points), approximation theory, numerical methods, coding theory, and control theory. Software systems for symbolic computation manipulate polynomials to perform simplifications, factoring, and solving tasks. The study of polynomials often emphasizes structure, factorization, and the behavior of roots.