polynomisk

Polynomisk is an adjective used in mathematics, especially in Swedish and related languages, to describe things related to polynomials. In practice, it is applied to expressions, functions, equations, and properties that can be described by a polynomial.

A polynomial in one variable x is an expression of the form a0 + a1 x + a2 x^2 + ... + an x^n, where the coefficients ai are constants and n is a nonnegative integer. A polynomial in several variables consists of sums of terms with nonnegative integer powers of the variables, possibly multiplied together, such as a x^2 y + b x y^2. The degree of a polynomial is the highest total degree of its nonzero terms, and the leading coefficient is the coefficient of the term with that degree.

A polynomisk funktion (polynomial function) is a function defined by a polynomial, for example f(x) = 2x^3 − x^2 + 5. Polynomial functions are continuous everywhere and differentiable to any order. They have a finite number of zeros, counted with multiplicity, and over the complex numbers they have exactly as many roots as the degree of the polynomial (Fundamental Theorem of Algebra).

Operations on polynomials are closed under addition, subtraction, multiplication, and scalar multiplication, but division by a polynomial generally yields a non-polynomial expression. Polynomials are central in algebra and analysis, used in approximation (Taylor polynomials, interpolation), solving equations, and modeling a wide range of phenomena. The terminology “polynomisk” thus communicates a connection to polynomial structure or behavior.