For that matter, asking students at the high school level to derive the quadratic formula will likely be quite challenging for even the best students in the cla

For that matter, asking students at the high school level to derive the quadratic formula will likely be quite challenging for even the best students in the class. Asking students to come up with the formulas for 3rd or 4th degree polynomials on their own is probably going to be beyond anyone below the very top tier of human capability. A teacher would be extremely lucky to see one or two students that can do this in their career. Then again, most people could easily solve these problems if they know what terms to Google :) Anyway, as I recall the third- and fourth-degree cases require clever substitutions (whereas the derivation of QF is just completing the square with general coefficients).