A question about the definition of polynomials.

The subscript on the coefficients is just a way of labeling them. The meaning of "$a_5$" is, "the coefficient of the degree $5$ term".

For example, consider the third degree polynomial:

$$5x^3-11x^2 + 9$$

In this case, we have $n=3$, because the degree is $3$. The coefficients are: $a_3=5, a_2=-11, a_1=0, a_0=9$. Thus, the $a$ notation is just a clear way of referring to each of the coefficients.

Does that help?

In mathematics we often us $n$ to describe a template. The expression you've written is the form that all polynomials have (though you're missing the term $+a_0x^0$).

So $x^3-2x^2+0x+1$ is a polynomial with $n=3$ and $x+1$ is a polynomial with $n=1$. Often times we drop terms with a coefficient of $0$, but I've included it to make the template clearer.

The general form for $n=5$ is $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x^2+a_0x^0$.