If you’ve ever taken an algebra or physics class, then you’ve met a parabola, the simple curve that can model how a ball flies through the air. The most important part of a parabola is the vertex — its highest or lowest point — and there are many mathematical techniques for finding it. You can try vertex form, or the axis of symmetry, or even calculus.
But last week one of my students located the vertex of a parabola in a particularly elegant way. “The vertex is at x = 4,” she said, “because the roots are x = 1 and x = 7, and the roots are symmetric about the vertex.” She used the fact that the parabola is the graph of a quadratic polynomial, and that the roots of that polynomial — the values where it becomes 0 — have a certain structure she could take advantage of.
There is a structure to the roots of every polynomial, and mathematicians study these structures and look for opportunities to capitalize on them, just as my student did with her parabola. And when it comes to the roots of polynomials, none have more structure than the “roots of unity.”