quality:corruption
Definition

A polynomial expression is an expression that can be built from constants and symbols called variables or indeterminates by means of addition, multiplication and exponentiation to a non-negative integer power. The constants are generally numbers, but may be any expression that do not involve the indeterminates, and represent mathematical objects that can be added and multiplied. Two polynomial expressions are considered as defining the same polynomial if they may be transformed, one to the other, by applying the usual properties of commutativity, associativity and distributivity of addition and multiplication. For example (x − 1) (x − 2) {\displaystyle (x-1)(x-2)} and x 2 − 3 x + 2 {\displaystyle x^{2}-3x+2} are two polynomial expressions that represent the same polynomial; so, one has the equality (x − 1) (x − 2) = x 2 − 3 x + 2 {\displaystyle (x-1)(x-2)=x^{2}-3x+2}.

A polynomial in a single indeterminate x can always be written (or rewritten) in the form a n x n + a n − 1 x n − 1 + ⋯ + a 2 x 2 + a 1 x + a 0, {\displaystyle a_{n}x^{n}+a_{n-1}x^{n-1}+\dotsb +a_{2}x^{2}+a_{1}x+a_{0},} where a 0, …, a n {\displaystyle a_{0},\ldots,a_{n}} are constants that are called the coefficients of the polynomial, and x {\displaystyle x} is the indeterminate. The word "indeterminate" means that x {\displaystyle x} represents no particular value, although any value may be substituted for it. The mapping that associates the result of this substitution to the substituted value is a function, called a polynomial function.

This can be expressed more concisely by using summation notation: ∑ k = 0 n a k x k {\displaystyle \sum _{k=0}^{n}a_{k}x^{k}} That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero terms. Each term consists of the product of a number – called the coefficient of the term – and a finite number of indeterminates, raised to non-negative integer powers.