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Polynomial expression

A polynomial expression is an expression built with scalars (elements of K), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers.

As all these operations are defined in K {\displaystyle K} a polynomial expression represents a polynomial, that is an element of K. {\displaystyle K.} 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.