He's stumbled onto the symmetric polynomials from Newton. It's part of a deeply fascinating area of mathematics with an astounding number of non-trivial results.
Knowing stuff about them let's you get powers of polynomials easily, or know certain things about roots (is there a real one? What can you tell about the signs? Lots of neat identities for special kinds of matrices...)
The "coeffs from roots" stuff is pretty straightforward to prove.
"Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients" [1]