Seeking short algebraic proofs that an Algebra2 student can appreciate

Question

There are many many elegant algebraic (more generally, non-geometric) proofs, but fewer of them are both accessible and interesting to a pre-calculus student. Three nice examples of what I'm looking for are the proof of the irrationality of the square root of 2, Cantor's diagonal argument (although I don't know if that's considered an actual proof), and the proof of Euler's formula by algebraic manipulation of the expansions of exp, sin, and cos. I'm seeking a dozen or so more like these, i.e., mostly algebraic, elegant, fun (if you're a math geek :-), and short (you could write them down in ~1 page). Thanks in advance!

(Clarification: If the proof has geometric aspects, like the diagonal argument, that's fair game. What I want to avoid is primarily geometric proofs, like those you do in your first geometry course. It's not that I'm against those, it's just that those are really easy to find as there are books and books of them -- like, specifically, 13 of them! :-) Also, I'm trying to avoid "proofs without words", which I find personally annoying because, although they may not literally have words once you see how the proof works, getting there usually involves a LOT of explanation! So the label "proofs without words" is somewhat disingenuous to my mind.)

Answer 1

A lot of the "pre-calculus curriculum" (at least in my day) would seem to qualify.

I remember my textbook containing proofs of:

1. The quotient-remainder theorem for real polynomials.
2. The corollary that roots are in correspondence with linear factors.
3. The corollary that a polynomial of degree $n$ can have at most $n$ roots.
4. The rational root theorem.
5. The derivation of the quadratic formula, and the use of discriminants to distinguish reducible from irreducible quadratics.
6. The binomial theorem.
7. A lot of interesting results in analytic geometry concerning conic sections (how to complete the square with two variable quadratics, how to rotate a conic section into "standard form", how to compute the location of foci and directrixes, how to compute eccentricities, etc).

The course I took did not expect me to learn or reproduce these proofs: only utilize (some of) the results. I have fond memories of learning this stuff anyways (I have always been a completionist: I wanted to master the whole book).

I think it is worth remembering that even if these "basic" things have lost their luster for you, a high school student is meeting these for the first time. If introduced with care, to the right students, proving these results can be a profound mathematical experience.

Answer 2

My student the other day was delighted with my "proof" that the perpendicular to a line of gradient $m$ has gradient $-1/m$, which I showed him by drawing the standard rise-over-run triangle for one line, drawing a perpendicular to that line, then turning the paper $90^\circ$ and drawing the rise-over-run triangle for that, pointing out that it's the same triangle as the first one I drew, then turning it back the right way and showing that the rise-over-run of the second line had magically turned into minus the run over the rise of the first. He loved it.