Two Questions Regarding Polynomial Functions. I am studying Polynomial functions and their Graphs.
I am currently looking at the definition for a polynomial function and I am trying to arrive at a deeper understanding; thus, please excuse questions that seem obvious. 
Nonetheless, 
A polynomial function of degree $n$ (What is $n$, a arbitrary variable chosen?) is a function of the form: 
$$
P(x) = a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_1 x + a_0
$$
where $n$ is a nonnegative integer and $a_n$ does not equal 0.
The numbers $a_0$, $a_1$, $a_2$, $\ldots$, $a_n$ are called the coefficients of the polynomial. 
The number $a_0$ is the constant coefficient or constant term. 
The number $a_n$, the coefficient of the highest power, is the leading coefficient, and the term $a_n x^n$ is the leading term. 
Questions:
1) What do the subscripts indicate, such as $n$?
2) In algebra, I learned that constants are for example $1, 2, 3, 4$; however, they describe constants as constant coefficients in the book. Can someone explain the reason behind that? 
As you can see, I am new to learning mathematics, so please be simple. 
 A: Here is a concrete example which you can compare to the definition to perhaps gain some understanding. 
The following expression is a polynomial of degree 5,
$$ 3x^5-2x^4+x^2-x+2.$$
When I say it has a degree of 5 all I mean is that the highest power of $x$ which you will find in the expression is the 5th power.
The coefficients in this polynomial are 3,-2,0,1,-1, and 2. Since each power of $x$ has its own coefficient it is sometimes useful to give them the following labels.
$$ a_5 = 3$$
$$ a_4 = -2$$
$$ a_3 = 0$$
$$ a_2 = 1$$
$$ a_1 = -1$$
$$ a_0 = 2$$
Notice that each coefficient has as a subscript the power of $x$ that the coefficient is multiplied. This is just a notational convention for labeling or organizing coefficients and it isn't really mathematically important, but it is useful to understand. 
It may seem odd to you that I have a coefficient $a_3=0$ listed. This is just there to indicate that our polynomial doesn't have an $x^3$ term.
A: It is just to make the writing more general. As you wrote $$P_n(x)=a_n x^n+a_{n-1} x^ {n-1}+a_{n-2} x^{n-2}+...+a_{2} x^2+a_1 x+a_0$$ The subscript $k$ assigned to the coefficient is the one associated to power $k$ of the variable. You will see soon that, because of the generalization, we can shorten the writing to $$P_n(x)=\sum _{k=0}^n a_k x^k$$ 
When you will have a practical problem, the coefficients $a_k$ will be replaced by numbers.
Is this making things clearer to you ? If not, please post.
A: 1) The subscripts are an easy and clever way to denote the coefficients of every term. For example, one could write a polynomial function as: $$f(x)=\alpha x^n+\beta x^{n-1}+\gamma x^{n-2}+\zeta x^{n-3}+\xi x^{n-4}+\cdots+\eta$$
But if I ask you for example: what is the coefficient of the term of degree $n-5$, you'll have to search and look for the symbol and recognize its name... It also happens that we don't have infinitely many symbols so you'll have a hard time writing the standard form of a polynomial with coefficient $200$ for instance. That's why the $a_{\displaystyle\color{grey}{\text{subscript}}}$ notation was adopted. So when you write the standard form and I ask you to find the coefficient of the term with degree $n-30$ you'll easily respond: $a_{n-30}$.
2) Why we say that they are constant coefficients is that you can view them as a constant times a variable: $$f(x)=a^nx^n+\cdots+a_1x+\color{blue}{a_0x^0}.$$
I hope this helps. 
Best wishes, $\mathcal H$akim.
