polynomia

Polynomia is a term occasionally used to refer to a polynomial, a fundamental object in algebra. In modern mathematics, the standard term is polynomial. A polynomial in one variable x over a ring R is an expression of the form a0 + a1 x + a2 x^2 + ... + an x^n, where the coefficients ai belong to R and an ≠ 0. The degree of the polynomial is n; the zero polynomial is typically considered to have no meaningful degree.

Polynomials can be extended to multiple variables, using monomials in several variables such as x^i y^j ..., producing multivariate polynomials like f(x,y) = a00 + a10 x + a01 y + a11 xy + ... . The collection of polynomials with coefficients in a fixed ring forms a polynomial ring, denoted R[x] or R[x1,...,xm].

Operations on polynomials include addition, multiplication, and division with remainder, as well as composition. Central concepts include roots or zeros, factorization into irreducibles, and the Factor Theorem: a value r is a root of f if and only if (x − r) divides f. Over the complex numbers, the Fundamental Theorem of Algebra asserts that every nonconstant polynomial has a complex root; over the real numbers, nonreal roots occur in conjugate pairs.

Polynomials have broad applications in mathematics and applied fields, including solving equations, interpolation and approximation (such as polynomial interpolation and Taylor series), numerical methods, coding theory, and algebraic geometry. They are studied within the framework of polynomial rings and their algebraic and analytic properties.