Consider a quadratic function f(x) Let m be the average of the roots then the point (m,f(m)) is a stationary point Cool! Now consider a cubic function g(x) Let n be the average of the roots Where is the point (n,g(n))? What do you notice?

Including the complex roots, of course...

Investigating the cubic case is lovely exercise for A-Level pupils. I wonder how A-Level teachers would know the result?