Editing Polynomial (section)

== Definition ==
A ''polynomial expression'' is an [[expression (mathematics)|expression]] that can be built from [[constant (mathematics)|constants]] and symbols called ''variables'' or ''indeterminates'' by means of [[addition]], [[multiplication]] and [[exponentiation]] to a [[non-negative integer]] power. The constants are generally [[number]]s, but may be any expression that do not involve the indeterminates, and represent [[mathematical object]]s 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 [[commutative property|commutativity]], [[associative property|associativity]] and [[distributive property|distributivity]] of addition and multiplication. For example <math>(x-1)(x-2)</math> and <math>x^2-3x+2</math> are two polynomial expressions that represent the same polynomial; so, one has the [[equality (mathematics)|equality]] <math>(x-1)(x-2)=x^2-3x+2</math>.

A polynomial in a single indeterminate {{math|''x''}} can always be written (or rewritten) in the form
<math display="block">a_n x^n + a_{n-1}x^{n-1} + \dotsb + a_2 x^2 + a_1 x + a_0,</math>
where <math>a_0, \ldots, a_n</math> are constants that are called the ''coefficients'' of the polynomial, and <math>x</math> is the indeterminate.<ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Polynomial|url=https://mathworld.wolfram.com/Polynomial.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> The word "indeterminate" means that <math>x</math> 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 (mathematics)|function]], called a ''polynomial function''.

This can be expressed more concisely by using [[summation#Capital-sigma notation|summation notation]]:
<math display="block">\sum_{k=0}^n a_k x^k</math>
That is, a polynomial can either be zero or can be written as the sum of a finite number of non-zero [[Summand|terms]]. Each term consists of the product of a number{{snd}} called the [[coefficient]] of the term{{efn|The coefficient of a term may be any number from a specified set. If that set is the set of real numbers, we speak of "polynomials over the reals". Other common kinds of polynomials are polynomials with integer coefficients, polynomials with complex coefficients, and polynomials with coefficients that are integers [[modular arithmetic|modulo]] some [[prime number]] {{math|''p''}}.}}{{snd}} and a finite number of indeterminates, raised to non-negative integer powers.