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 exponentiation 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 many 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 scientific problems; 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, which are central concepts in algebra and algebraic geometry.

Thus the set of all polynomials with coefficients in the ring *R* forms itself a ring, the *ring of polynomials* over *R*, which is denoted by *R*[*x*]. The map from *R* to *R*[*x*] sending *r* to *rx*^{0} is an injective homomorphism of rings, by which *R* is viewed as a subring of Thus the set of all polynomials with coefficients in the ring *R* forms itself a ring, the *ring of polynomials* over *R*, which is denoted by *R*[*x*]. The map from *R* to *R*[*x*] sending *r* to *rx*^{0} is an injective homomorphism of rings, by which *R* is viewed as a subring of *R*[*x*]. If *R* is commutative, then *R*[*x*] is an algebra over *R*.

One can think of the ring *R*[*x*] as arising from *R* by adding one new element *x* to *R*, and extending in a minimal way to a ring in which *x* satisfies no other relations than the obligatory ones, plus commutation with all elements of *R* (that is *xr* = *rx*). To do this, one must add all powers of *x* and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers, which can be constructed from the polynomial ring *R*[*x*] over the real numbers by factoring out the ideal of multiples of the polynomial *x*^{2} + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring *R* (see modular arithmetic).

If *R* is commutative, then one can associate with every polynomial