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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., $a:b:c$?

147pm
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  • Just imagine them as "over 1", and they will be in ratio with $\Huge 1$. – Kenny Lau May 08 '16 at 15:36
  • @ Kenny Lau Yes, that much I understand. It's the "ratio" part I don't understand. Is there a really general, theoretic theorem of "ratio" that would encompass my volume example? – 147pm May 08 '16 at 16:01
  • Do you believe that an integer is also a rational number? If so, what's different about polynomials? If not, you should probably start there... – Micah May 08 '16 at 16:11
  • The constant function 1 is a polynomial function. – DanielWainfleet May 08 '16 at 18:31

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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.

Harry Altman
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  • @ Harry Altman Thank you. But are you then saying the "ratio" aspect of this volume example is simply $p/1$, or $[p : 1]$? Again, my main concern is how I can see this example in terms of a ratio. As such, calling it $p/1$ hardly brings much "ratio" to the table. – 147pm May 08 '16 at 16:55
  • Yes. It's a rational function because it's a polynomial; it's $p/1$. There's nothing hidden going on here. It really is that simple. Whether it calls to mind a ratio or not is irrelevant to whether it is one. – Harry Altman May 08 '16 at 17:31
  • Solving this, we get $V = (x + 3)^2\pi(x - 6)$. It does seem, since we've got the variable $x$ that this is a ratio. As another example (from a high school text), consider the volume of a rectangular prism: $6x^3+11x^2+4x$, which is $x(2x+1)(3x+4)$. Then it says "Find the ratio of the three dimensions when $x=2$". Then, "Will the ratio of the three dimesions be the same for all $x$?" The answer is no, of course. So, yes, this speaks in terms of ratio. But you're saying this is just coincidence? – 147pm May 08 '16 at 18:07
  • Yes, this is a coincidence. If you already know the sides of a box, finding its volume and finding the ratios of the sides don't have much to do with one another. It does not phrase the problem in terms of ratio; it presents an entirely different problem, which is about ratio. – Harry Altman May 08 '16 at 19:12
  • Although, both of these problems are unsolvable as written. If you only know the volume of a cylinder, you can't determine its height; if you only know the volume of a box, you can't determine its dimensions. The writers of these problems seem to be assuming that the easily visible factors are the dimensions. But mathematically there's no reason that needs to be true. So although I said earlier that there's nothing hidden going on, I guess there actually is a hidden assumption being made; that's what happens when you have poorly written problems. But there's still no hidden rational function. – Harry Altman May 08 '16 at 19:13