Occasionally, in contest problems, it helps to have a careful understanding of real numbers and real-valued functions. But what, exactly, is a real number? These Auckland squad traininglecture notesby Heather Macbeth outline some basics.
Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors. Why? Well, here’s a polynomial, see. It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are. You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients. Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding. Tada!
Of course, there’s a problem with this algorithm: it depends on figuring out how to break down your polynomial into only linear and quadratic factors.
These notesby Arkadii Slinko explain how to extract information from symmetric polynomials of a set of variables, and how to break any symmetric polynomial down into a few simple ones. The final section gives some applications to triangle geometry.