Polünoomne

Polünoomne is a mathematical concept used to describe an expression formed by a finite sum of terms a_i x^i where i are nonnegative integers and a_i belong to a given ring or field. Equivalently, P(x) = a_n x^n + ... + a_1 x + a_0 with a_n ≠ 0 is called a polynomial of degree n. The set of all polynomials with coefficients from a given ring R is denoted R[x], and forms a ring under the usual addition and multiplication of polynomials.

Polynomials can be evaluated at elements of the coefficient ring by substituting the variable with a value: P(r) where r ∈ R. When evaluated for every r in R, this yields a polynomial function. A polynomial is a formal expression, while a polynomial function is the actual function defined by evaluation; in general they coincide over infinite rings but may differ over finite rings.

Common types include constant polynomials (degree 0), monomials a x^n, linear polynomials (n = 1), and quadratic polynomials (n = 2). The degree, given by the highest power with a nonzero coefficient, controls many properties of P.

Key properties include factorization: a polynomial can often be written as a product of irreducible polynomials. Over algebraically closed fields such as the complex numbers, every nonconstant polynomial factors completely into linear factors. Over the real numbers, factors may be linear or irreducible quadratics.

Applications span solving polynomial equations, interpolation and approximation (for example, Taylor polynomials), generating functions, coding theory, and computer algebra. Polynomials also serve as building blocks for more complex objects in algebra and analysis.