Why is a polynomial also a rational function? In a recent question, I asked about non-standard-looking rational functions, i.e., something that was not in the classic numerator-denominator form. I was told that all polynomials are rational functions, that perhaps I should just imagine them as "over 1". Good. But the definition of a rational function has the concept of "ratio" in it. And when I found this:

A cylinder has a volume of $(x+3)(x^2-3x-18)\pi$ cubic centimeters. Find the height of the cylinder.

I wondered how this is a ratio of any sort? (It can't be a ratio of the expression over 1, is it?) So if $a = (x+3)$, $b = (x^2-3x-18)$, and $c = \pi$, then can we say $a, b, c$ are "in a ratio," i.e., $a:b:c$?

You seem to be asking multiple different questions that are not actually related to each other.
A polynomial $p$ is a rational functions for the reason that's already been said; it's $p/1$.
You seem to be trying to extract something else from this cylinder example; something about the the three factors being in some specific ratio with each other.  There is no such thing in the problem.  The factors are what they are; they are not some other thing with the same ratio.  All that's going on here is that you have a polynomial.
That said, there is a notion in mathematics of a "ratio" between more than two things -- $a:b:c$, as you say.  This is what is called projective space.  In math the usual notation is to put brackets around it -- $[a:b:c]$.  If you want to learn about ratios between more than two quantities, that's the appropriate notion.  That said, this has no relation to the cylinder problem.
