Are Polynomials more than just functions? I have been wondering for a while that polynomials aren't just functions they are more than that. I think it's quite convincing to think of polynomials as functions adjoined with a finite sequence in some field $\mathbb{F}$. For instance, I would consider the ordered pair  $(p,\{a_{n}\}_{n=0}^{N})$ to be a polynomial where p is a function and $p(x)=\sum_{n=0}^{N}a_nx^n$ for all $x$ in $\mathbb{F}$. The reason behind doing this is that I can think of distinct polynomials that represent the same function. Am I thinking in the right direction? 
 A: Yes, for example if the coefficients are from the ring $\mathbb{Z}_P$, $p$ a prime, to the two polynomials $X$ and $X^p$ is associated  the same polynomial function.
A: Basically, polynomials are not functions: they're sequences of coefficients $(a_0, a_1, a_2,\dots, a_n,\dots)$ from a commutative ring $R$, with finite support, i.e. such that only a finite number is nonzero. The set of such sequences is denoted $R^{(\mathbf N)}$ and is endowed with a componentwise addition, and scalar multiplication, which makes it an $R$-module. Furthermore, a product $(a_n)_{n\in\mathbf N}\cdot(b_n)_{n\in\mathbf N}=(c_n)_{n\in\mathbf N}$ is defined, where
$$\forall n\in\mathbf N,\quad c_n=\sum_{i+j=n}a_ib_j.$$
It happens that, if we denote $X$ the particular sequence $(0,1,0,\dots,0,\dots)$, each polynomial $P=(a_0, a_1, \dots, a_d, 0,\dots,0,\dots)$, where $a_d $ is the last nonzero coefficient, we can write
$$P=a_0+a_1X+\dots+a_d X^d,$$
in a unique way.
To this polynomial is associated  a polynomial function
\begin{align}
p:R&\longrightarrow R \\
r&\longmapsto p(r)=a_0+a_1r+\dots+a_dr^d,
\end{align}
and this correspondence is bijective if the ring $R$ has characteristic $0$.
A: In an infinite field there exists a bijection between the polynomial and polynomial function so one can regard polynomials as functions but unfortunately that doesn't happens in arbitrary fields (Friedberg Linear Algebra 4th editions explains that in page 569). Consider for instance $Z_2$.
