Polynomiel

A polynomiel, or polynomial, is a mathematical expression built as a finite sum of terms of the form a k x^k, where x is a variable and the coefficients a k come from a given ring or field. It is commonly written as P(x) = a_n x^n + ... + a_1 x + a_0, with a_n ≠ 0. The highest exponent n is called the degree. The all-zero coefficients yield the zero polynomial, whose degree is undefined or sometimes defined as negative infinity. Polynomials can be defined with coefficients in any ring, and in particular over the real or complex numbers.

Polynomials support standard algebraic operations: addition, subtraction, and multiplication are performed by combining like terms. There is a division algorithm: for any nonzero divisor G, there exist unique polynomials Q and R such that F = QG + R, with either R = 0 or deg(R) < deg(G). Evaluating a polynomial at a point c gives a polynomial function P(c). The remainder theorem states that the remainder of P when divided by (x − c) equals P(c).

Roots and factorization are central topics. A number r with P(r) = 0 is a root of P. The Fundamental Theorem of Algebra states that over the complex numbers a non-constant polynomial of degree n has exactly n roots counted with multiplicities. Over the reals, nonreal roots occur in conjugate pairs, and polynomials can be factored into irreducible linear and quadratic factors.

Polynomials form the polynomial ring R[x] over a ring R. They appear in approximations (Taylor polynomials), interpolation (Lagrange, Newton), and various computational methods in algebra and numerical analysis.