Now that you’ve got a product equal to 0, you can invoke one of the most underappreciated rules from algebra class: the “zero product property.” This says that the only way two real numbers can multiply to 0 is for one of them to be 0. Here x 2 – 1 2 = ( x – 1)( x + 1), which gives you You might remember factoring expressions like this using the “difference of squares” formula, which says that a 2 – b 2 = ( a – b)( a + b). To find its roots, just set it equal to 0 and solve: Roots of unity are the roots of the polynomials of the form x n – 1. For example, when n = 2, this gives us the quadratic polynomial x 2 – 1. And when it comes to the roots of polynomials, none have more structure than the “roots of unity.” 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. “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. You can try vertex form, or the axis of symmetry, or even calculus.īut last week one of my students located the vertex of a parabola in a particularly elegant way.

The most important part of a parabola is the vertex - its highest or lowest point - and there are many mathematical techniques for finding it. 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.

Quadratic polynomial example