polynomille

Polynomials are mathematical expressions built from variables and coefficients using addition, subtraction, and multiplication, with nonnegative integer exponents of the variable. In one variable x, a polynomial of degree n is written as P(x) = a0 + a1 x + a2 x^2 + ... + an x^n, where an ≠ 0. The degree is n and the leading coefficient is an. Coefficients come from a chosen field or ring, such as the real numbers, the complex numbers, or the integers.

Polynomials in several variables are sums of terms with multiple variables, for example P(x, y) = c00 + c10 x + c01 y + c11 xy + ... , with total degree determined by the largest sum of exponents in a term.

Polynomials are closed under addition, subtraction, and multiplication. Division by a nonzero polynomial gives a quotient and remainder; the division algorithm yields a unique remainder with degree less than the divisor. The remainder theorem and factor theorem connect roots to factors: f(c) = 0 implies (x − c) is a factor of f.

The Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficients factors completely into linear factors over the complex numbers. Real polynomials either factor into real linear factors or into irreducible quadratic factors.

Applications include solving equations, curve sketching, and numerical methods. Polynomials serve in interpolation (Lagrange and Newton forms), Taylor approximations, and as basis functions in approximation theory and computer algebra systems.

In Finnish and some other languages, the word for polynomial has inflected forms such as polynomille, reflecting grammatical case rather than changing the mathematical concept.