polynomials

A polynomial is an expression formed from variables and coefficients using only addition, subtraction, and multiplication, with nonnegative integer exponents of the variable. In one variable x, a polynomial of degree n is written in standard form as P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where the coefficients a_i come from a field (often the real or complex numbers) and a_n ≠ 0. The set of nonzero terms is finite.

The degree of a polynomial is the largest exponent with a nonzero coefficient; the zero polynomial has no defined degree (some sources assign −∞). The coefficients are elements of a ring or field, commonly R or C. Polynomials form an algebra over the chosen coefficient field, closed under addition and multiplication.

Evaluating a polynomial at a real or complex value yields a polynomial function. A root or zero is a value x with P(x) = 0. Roots may be repeated; the multiplicity counts how many times a root occurs in the factorization. Over the complex numbers, every nonconstant polynomial factors into linear terms according to the Fundamental Theorem of Algebra; over the real numbers, factors may be linear or quadratic irreducibles.

Key tools include the remainder theorem and factor theorem: the remainder of P divided by x − a equals P(a); x − a is a factor of P iff P(a) = 0. Polynomial division yields a quotient and remainder. Vieta's formulas relate sums and products of roots to coefficients. Polynomials are used in approximation, interpolation, solving equations, and modeling curves.