Polynomi

Polynomi is a mathematical expression consisting of a finite sum of terms, each term a coefficient times a variable raised to a nonnegative integer power. Coefficients typically come from a field such as the real or complex numbers, but may be taken from other algebraic structures as well.

In one variable, a Polynomi has the form p(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 with a_n ≠ 0. The degree of p is n, and the leading coefficient is a_n. In several variables, a Polynomi is a sum of monomials c_{i,j,...} x^i y^j z^k with nonnegative exponents; the total degree is the maximum value of i+j+... among nonzero terms.

Operations on polynomials include addition, subtraction, and multiplication. They can be evaluated by substituting numbers for the variables, and they can be factored into products of polynomials. Zeros or roots are the solutions to p(x) = 0 (and, for multivariable polynomials, the common zeros of the system p(x,y,...)=0). Over the complex numbers, the Fundamental Theorem of Algebra states that a univariate Polynomi of degree n has exactly n roots counted with multiplicity.

Common types encompass univariate and multivariate polynomials, as well as linear, quadratic, cubic, and quartic polynomials, and constant polynomials. Polynomials are foundational in algebra and geometry and have wide applications in numerical analysis, modeling, interpolation, and approximation, including techniques such as Taylor series and polynomial regression.