Editing Polynomial (section)

== Notation and terminology ==
[[File:Polynomialdeg3.svg|The [[graph of a function|graph]] of a polynomial function of degree 3|thumb]]

The ''x'' occurring in a polynomial is commonly called a ''variable'' or an ''indeterminate''. When the polynomial is considered as an expression, ''x'' is a fixed symbol which does not have any value (its value is "indeterminate"). However, when one considers the [[function (mathematics)|function]] defined by the polynomial, then ''x'' represents the argument of the function, and is therefore called a "variable". Many authors use these two words interchangeably.

A polynomial ''P'' in the indeterminate ''x'' is commonly denoted either as ''P'' or as ''P''(''x''). Formally, the name of the polynomial is ''P'', not ''P''(''x''), but the use of the [[functional notation]] ''P''(''x'') dates from a time when the distinction between a polynomial and the associated function was unclear. Moreover, the functional notation is often useful for specifying, in a single phrase, a polynomial and its indeterminate. For example, "let ''P''(''x'') be a polynomial" is a shorthand for "let ''P'' be a polynomial in the indeterminate ''x''". On the other hand, when it is not necessary to emphasize the name of the indeterminate, many formulas are much simpler and easier to read if the name(s) of the indeterminate(s) do not appear at each occurrence of the polynomial.

The ambiguity of having two notations for a single mathematical object may be formally resolved by considering the general meaning of the functional notation for polynomials.
If ''a'' denotes a number, a variable, another polynomial, or, more generally, any expression, then ''P''(''a'') denotes, by convention, the result of substituting ''a'' for ''x'' in ''P''. Thus, the polynomial ''P'' defines the function
<math display="block">a\mapsto P(a),</math>
which is the ''polynomial function'' associated to ''P''.
Frequently, when using this notation, one supposes that ''a'' is a number. However, one may use it over any domain where addition and multiplication are defined (that is, any [[ring (mathematics)|ring]]). In particular, if ''a'' is a polynomial then ''P''(''a'') is also a polynomial.
 
More specifically, when ''a'' is the indeterminate ''x'', then the [[Image (mathematics)|image]] of ''x'' by this function is the polynomial ''P'' itself (substituting ''x'' for ''x'' does not change anything). In other words,
<math display="block">P(x)=P,</math>
which justifies formally the existence of two notations for the same polynomial.