Polynome

Polynome, in mathematics, is an expression formed by a finite sum of terms, each consisting of a coefficient multiplied by a nonnegative integer power of a variable. When the coefficients come from a field or ring, a polynomial is a formal object that can be added, subtracted, multiplied, and, in suitable settings, divided with remainder. A common example in one variable is P(x) = a_n x^n + ... + a_1 x + a_0 with a_n ≠ 0. For several variables, polynomials are sums of monomials like x^i y^j with nonnegative integer exponents and coefficients.

The degree of a polynomial in one variable is the largest exponent with a nonzero coefficient. In several variables, the degree can be defined in several ways, such as the total degree or partial degrees. A zero polynomial has all coefficients equal to zero, and a monomial is a single term a x^n. Polynomials can be factored into irreducible polynomials, and over a field they satisfy a division algorithm: given P and Q with Q ≠ 0, there exist unique polynomials S and R such that P = Q S + R and either R = 0 or deg(R) < deg(Q).

Roots of polynomials are the values that make the polynomial equal to zero. The Fundamental Theorem of Algebra states that every nonconstant polynomial with complex coefficients factors into linear factors, counting multiplicities. Over the real numbers, complex roots occur in conjugate pairs.

Polynomials have wide applications in algebra, numerical analysis, physics, computer science, and coding theory, including interpolation, approximation, solving equations, and constructing algebraic objects. The term derives from Greek roots meaning “many parts.”