The average of the roots of a polynomial equals the average of the roots of its derivative
Solution 1:
I would check if this (i.e. sum of roots being $a_{n-1}$ up to sign) is in Simon Stevin (1548 – 1620). He is credited with some remarkable accomplishments including a proof of the intermediate value theorem in the context of a certain 3rd degree polynomial, as discussed in this article. This is all the more remarkable since he did not have any symbolic notation at all but rather expressed everything in terms of proportions, sometimes artificially so. He may well have been aware of some of the elementary facts for low-degree polynomials.