Polynomiellen

Polynomiellen is the inflected adjectival form found in German and related languages for the concept of polynomial. In mathematics, a polynomial is an algebraic expression formed by sums of terms, each consisting of a coefficient multiplied by a non‑negative integer power of a variable.

A single‑variable polynomial p(x) of degree n can be written as p(x) = a0 + a1 x + ... + an x^n, with an ≠ 0. Coefficients ai typically come from a ring or field, with real or complex numbers being common choices. Polynomials can be added, subtracted, multiplied, and composed, and they satisfy a division algorithm: for any nonzero divisor d, there exist polynomials q and r such that p = q d + r, with either r = 0 or deg r < deg d. The remainder theorem states that p(c) equals the remainder when p is divided by (x − c).

The Fundamental Theorem of Algebra asserts that every non‑constant polynomial with complex coefficients has exactly as many roots in the complex numbers as its degree, counted with multiplicities. Over the real numbers, roots are either real or occur in complex conjugate pairs. Multivariate polynomials involve several variables and form polynomial rings R[x1, x2, ..., xk] over a coefficient ring R.

Polynomials play a central role in many areas of mathematics and its applications, including interpolation, approximation, numerical analysis, coding theory, and algebraic geometry.