Definition:Polynomial Function/Real/Definition 1

Definition
Let $S \subset \R$ be a subset of the real numbers.

A real polynomial function on $S$ is a function $f : S \to \R$ for which there exist: such that for all $x\in S$:
 * a natural number $n\in \N$
 * real numbers $a_0, \ldots, a_n \in \R$
 * $f(x) = \displaystyle \sum_{k \mathop = 0}^n a_k x^k$

where $\sum$ denotes indexed summation.

Also see

 * Equivalence of Definitions of Real Polynomial Function