polys

Polys, short for polynomials, are algebraic expressions built from variables and constants by addition, subtraction, and multiplication, with nonnegative integer exponents of the variables. In one variable x, a polynomial p(x) has the form a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0, where a_n ≠ 0. The degree of p is n, and a_n is the leading coefficient. Coefficients are drawn from a ring or field, commonly the integers, real numbers, or complex numbers. Polynomials can also be defined in several variables, p(x1, x2, ..., xk), with each term a coefficient times a product of the variables raised to nonnegative integers.

Polynomials support standard operations such as addition, subtraction, and multiplication, and, with remainder, division by another polynomial. They can be evaluated at a given value by substituting numbers for the variables. Factorization is the process of expressing a polynomial as a product of irreducible polynomials over a chosen coefficient set, when possible. The Fundamental Theorem of Algebra implies that every nonconstant polynomial with complex coefficients factors completely into linear factors over the complex numbers; over the real numbers, the factors are linear and irreducible quadratic polynomials.

Common types include linear, quadratic, cubic, and higher-degree polynomials. Polynomials are central in algebra and analysis, serving as functions, as building blocks for polynomial rings, and as tools in numerical methods, approximation theory (Taylor polynomials), interpolation, and coding theory. Computationally, polynomials are manipulated symbolically in computer algebra systems, using algorithms for division, greatest common divisors, and factorization.