In mathematics, a **polynomial** is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate *x* is *x*^{2} − 4*x* + 7. An example in three variables is *x*^{3} + 2*xyz*^{2} − *yz* + 1.

Polynomials appear in a wide variety of areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define **polynomial functions**, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

The *x* occurring in a polynomial is commonly called either 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”). It is thus more correct to call it an “indeterminate”.^{[citation needed]} However, when one considers the 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.

It is a common convention to use uppercase letters for the indeterminates and the corresponding lowercase letters for the variables (arguments) of the associated function.^{[citation needed]}

It may be confusing that a polynomial *P* in the indeterminate *x* may appear in the formulas either as *P* or as *P*(*x*).^{[citation needed]}

Normally, the name of the polynomial is *P*, not *P*(*x*). However, if *a* denotes a number, a variable, another polynomial, or, more generally any expression, then *P*(*a*) denotes, by convention, the result of substituting *x* by *a* in *P*. Thus, the polynomial *P* defines the function

- {\displaystyle a\mapsto P(a),}

which is the polynomial function associated to *P*.

Frequently, when using this function, one supposes that *a* is a number. However one may use it over any domain where addition and multiplication are defined (any ring). In particular, when *a* is the indeterminate *x*, then the image of *x* by this function is the polynomial *P* itself (substituting *x* to *x* does not change anything). In other words,

- {\displaystyle P(x)=P.}

This equality allows writing “let *P*(*x*) be a polynomial” as 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.