A question about the definition of polynomials.

A part of the definition of a polynomial is :

$f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots+a_2x^2+a_1x^1+a_0$

where $a_n ,a_{n-1}, \dots, a_2, a_1, a_0$ are constants.

$\textbf{I have been confused as to why we have n.So if n is 5 do we have the following:}$ $$a_5x^5+a_4x^4+a_1x^1+a_0x^0$$

Please can I have help in understanding this ?

What in the world is $a_nx^n$ ?

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 $a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x^2+a_0x^0$.