Editing Polynomial ring (section)

===Polynomial expression===
{{main|Algebraic expression}}
{{Unreferenced section|date=January 2021}}
A '''polynomial expression''' is an [[expression (mathematics)|expression]] built with scalars (elements of {{mvar|K}}), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in <math>K[X_1,\dots, X_n]</math> a polynomial expression represents a polynomial, that is an element of <math>K[X_1,\dots, X_n].</math> The definition of a polynomial as a linear combination of monomials is a particular polynomial expression, which is often called the ''canonical form'', ''normal form'', or ''expanded form'' of the polynomial. 
Given a polynomial expression, one can compute the ''expanded'' form of the represented polynomial by ''expanding'' with the [[distributive law]] all the products that have a sum among their factors, and then using [[commutativity]] (except for the product of two scalars), and [[associativity]] for transforming the terms of the resulting sum into products of a scalar and a monomial; then one gets the canonical form by regrouping the [[like terms]].

The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of [[computer algebra]], where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.