500 years of NOT teaching THE CUBIC FORMULA. What is it they think you can't handle?

So instead of a second video, here are just three fun facts I stumbled upon while working on this video, without explanation. Don't hesitate to try them in the comments. Fun fact number one. Here is a

cubic

with three zeros. Draw the tangents at those zeros. The tangents intersect the graph at one more point each. So those three points are on a straight line. Always! Nice, huh? Fun fact two. These are again

cubic

s with three zeros. Highlight the turning point. Draw an equilateral triangle whose center is somewhere above the inflection point. Then it is always possible to rotate and scale this triangle so that all three corners end up above the zeros.

Good looking. You can easily prove this based on

what

I said in the previous chapter. But there is more. Inscribe a circle in the triangle. The two ends are then aligned with the leftmost and rightmost points of the circle like this. Final related fun fact and this one is a real killer: start again with a cubic polynomial, but this time the coefficients can be any complex number. So, in general, the three zeros of said polynomial will be three points in the complex plane, forming a triangle. Highlight the midpoints of the edges of this triangle. It turns out that there is exactly one ellipse that touches the edges at those midpoints.

And here's fun fact number three: the two zeros of the derivative of our polynomial are the focal points of this ellipse. This surprising result is called Marden's theorem. And one more thing is true. The zero of the second derivative, which is also the z coordinate of the "complex inflection point" is the center of the ellipse. And, phew, that's all for today, to finish I'll just show you an animation of a method for solving quadratic equations. This solution is due to Ludovico Ferrari's assistant Cadano. One of the super surprising and famous results in algebra is that Ferrari's method is the furthest we can go when asking for solutions in radicals.

There is no such quintic

formula

nor a

formula

for anything beyond that. As I said, demonstrating this will be the mission of a future Mathologer video. For now enjoy Ferrari and the animations and music. Until next time :)