Suppose $f$ is a real valued function with domain a subset of  real numbers, $D$.

Definition: $f$  is a polynomial function if there a numbers $a_0$, $a_1$, ... , and $a_n$   so that for any $x \in D$,  $f(x) = a_nx^n + ... a_2x^2 + a_1x +a_0 $. If $a_n \ne 0$ we say that $f$ is a polynomial of degree $n$.
Comment: The letters  $a_0$, $a_1$, ..., and $a_n$ have been used to conform with the naming of these coefficients in quadratic functions which are polynomials of degree $2$.

The number $a_n$ is sometimes referred to as  the "leading coefficient " of the polynomial function.
The number $a_1$ is sometimes referred to as  the "linear coefficient " of the polynomial function.
The number $a_0$ is sometimes referred to as "the $y$- intercept," "the initial value," or the "constant " of the polynomial function.