Reminiscing about a cute calculus trick
The year was probably 2008. I was a poorly dressed adolescent, with a burgeoning patchy mustache so awful that I still struggle with the shame of having had no desire to shave it off. (No photo posted intentionally, thank you very much.) I was attending a meeting of the Nassau County All-Star Math Team, which congregated some of the best high school students in the area where I grew up. As practice for upcoming math competitions ( NYSML and ARML ), we were given problems to work on (we had 10 minutes for every pair of problems), after which students would present their solutions. One such problem really sticks out in my mind, and I was reminded of it while teaching calculus today. I don't remember the exact problem, so here's a version suitable for 2022 (in a manner typical of those competitions). Take a few minutes to try to solve the problem, if you like! Problem: Let $p(x)$ be the polynomial satisfying $$x^{2022}-5x^{1011}+4 = (x-1) \cdot p(x).$$ Find the sum of the coeffi...
