Polynomial

A polynomial over a ring $R$ is a formal expression $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ where $a_i \in R$ and $x$ is an indeterminate. The set of all such expressions, with the usual addition and multiplication rules, forms the polynomial ring $R[x]$.

If $R$ is a commutative ring, so is $R[x]$. If $R$ is an integral domain, so is $R[x]$. Polynomial rings are the basic tool for constructing new rings from old ones and underpin algebraic geometry, where the zeros of polynomials define geometric objects.