Polünomiline

Polünomiline is a mathematical object typically defined as a finite expression formed from sums of nonnegative integer powers of a variable with constant coefficients. A common univariate form in one variable x is P(x) = a0 + a1 x + a2 x^2 + ... + an x^n, where n is a nonnegative integer and each coefficient ai comes from a chosen coefficient set, usually a field or a ring. If an ≠ 0, the degree of P is n; if all coefficients are zero, P is the zero polynomial and has no degree.

Polynomials can be added, subtracted, and multiplied to yield other polynomials, and they can be composed as well. They form a algebraic structure known as a ring under addition and multiplication, and when coefficients come from a field, this gives rise to a polynomial ring. Evaluation at a particular numeric value x0 produces a scalar from the coefficient set.

Zeros and factorization are central topics. A root of P is a value x0 with P(x0) = 0. Over the complex numbers, every non-constant polynomial can be factored into linear factors (the Fundamental Theorem of Algebra). Over other fields, factorization may involve irreducible polynomials of higher degree. The rational root theorem provides candidate rational roots for polynomials with integer coefficients.

Applications include modeling relationships, function approximation through Taylor or Maclaurin polynomials, and data interpolation using methods like Lagrange or Newton polynomials. Polynomials also appear in numerical methods for solving equations and in symbolic computation. Variants include multivariate polynomials in several variables and polynomial rings over various coefficient domains.