Background: I'm a teacher preparing a precalculus course for a group of mature students with wildly differing backgrounds, so I constantly have an eye out for more sophisticated topics to point the better-prepared ones towards, if only for interest. I especially like to at least acknowledge the existence of more advanced topics that unify or apply the ideas we're studying.
So I was writing up some notes about polynomials and noticed the rules for addition and multiplication of $f$ and $g$, where $\text{deg}(f)$ is the degree of $f$:
$$ \text{deg}(fg) = \text{deg}(f) + \text{deg}(g)\\ \text{deg}(f+g) = \text{max}( \text{deg}(f), \text{deg}(g) ) $$
I don't know any tropical geometry / algebra but this reminded me of the basic definitions I saw a long time ago when reading something above my pay grade:
$$ f\otimes g = f + g\\ f\oplus g = \text{max}(f, g) $$
From Wikipedia, my understanding is that the main definition uses $\text{min}$ but there's a variant that uses $\text{max}$ (and my guess is the resulting theories are just dual to each other?). But this seems way too similar to be a coincidence. And this makes me wonder whether a fruitful view of elementary properties of polynomials drops out of this way of looking at things without having to get too far into the weeds.
So, is this connection essentially just an accident or is is part of the historical motivation for the definition of the tropical semiring? Is there any part of this -- even a fun anecdote -- that might be meaningful to someone seeing the basic algebra of polynomials for the first time? (They will know roughly the difference between a ring and a field but not, for example, what an ideal is.)