Polynomial

A polynomial is an algebraic expression formed by adding together constants and variables raised to nonnegative integer powers, using only the operations of addition, subtraction, and multiplication. In one variable x, a polynomial with coefficients from a ring R is written as P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where n is a nonnegative integer and a_n ≠ 0. The degree of P is n, the leading coefficient is a_n, and the constant term is a_0. The zero polynomial, in which all coefficients are zero, is a special case and its degree is often defined as undefined or negative infinity.

For several variables, polynomials are finite sums of monomials a_{i1...ik} x1^{i1} ... xk^{ik} with nonnegative integer exponents. Coefficients are taken from a ring or field, commonly the real or complex numbers.

Operations on polynomials follow the usual algebraic rules: addition, subtraction, and multiplication, with scalar multiplication by elements of the coefficient ring. Division by polynomials is exact only in appropriate contexts, typically via polynomial long division or within a field of fractions.

Zeros and factorization: a number r is a root of P if P(r) = 0. Over algebraically closed fields, every non-constant polynomial factors into linear factors, counted with multiplicity (Fundamental Theorem of Algebra). Over the integers or rationals, factorization proceeds into irreducible polynomials.

Polynomial functions: when coefficients are real or complex, P defines a function from the real line to itself, continuous and differentiable, with graph shapes determined by the degree and coefficients. They play central roles in approximation, interpolation, solving equations, and modeling.