Is polynomial in general the same as polynomial function? The algebra text book says, a polynomial in one variable over $\mathbb{R}$ is given by, 
$$f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$$
Where $x$ is an unknown quantity which commutes with real numbers, called "indeterminate".
So I have a few question,


*

*Is polynomial always a function? If not then what is a polynomial in general?

*And what's up with the "indeterminate" thingy? Is it wrong to simply call it a variable? 

*What exactly is the $x$ in the expression? A number? A matrix? Or… some other object? Why does it have to "commute" with real numbers?


No, a polynomial is not a function. However, for each polynomial $p(x)=a_0+a_1x+\cdots+a_nx^n$ you may consider the polynomial function$$\begin{array}{rccc}p\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&a_0+a_1x+\cdots+a_nx^n.\end{array}$$And distinct polynomials will be associated with distinct functions. However, although this is true over the reals, it doesn't hold in general. For instance, if you are working over the field $\mathbb{F}_2$, then the polynomial $x^2-x$ and the null polynomial are distinct polynmials, but the function$$\begin{array}{ccc}\mathbb{F}_2&\longrightarrow&\mathbb{F}_2\\x&\mapsto&x^2-x\end{array}$$is the null function.
So, a polynomial (over the reals) is an expression of the type $a_0+a_1x+\cdots+a_nx^n$, where $x$ is an entity about which all we assume is that it commutes with each real number. Usually, it is called an “indeterminate” since it is not a specific real number.
