Vieta's Formulae and Formation of Polynomial Equations

Vieta's Formulae, named for the French mathematician François Viète, relate the coefficients of a polynomial equation to sums and products of its roots — without ever needing to know the individual roots themselves. They answer two closely related questions that recur throughout this chapter:

1. Given the roots of an equation, what equation has them? (Trivial once we know the sum/product-type combinations.)
2. Given an equation, what can we say about combinations of its roots (their sum, their pairwise products, $\sum 1/\alpha$, etc.) purely by reading off its coefficients — again, without solving it? …