Universal Peculiar Linear Mean Relationships in All Polynomials

In any cubic polynomial, the average of the slopes at the $3$ roots is the negation of the slope at the average of the roots. In any quartic, the average of the slopes at the $4$ roots is twice the negation of the slope at the average of the roots. We generalize such situations and present a procedure for determining all such relationships for polynomials of any degree. E.g., in any septic $f$, letting $\overline{f}_n$ denote the mean $f$ value over all zeroes of the derivative $f^{\left(n\right)}$, it holds that $37$ $\overline{f}_1-150$ $\overline{f}_3+200\,\overline{f}_4-135\,\overline{f}_5+48\,$; and in any quartic it holds that $5$ $\overline{f}_1-6$ $\overline{f}_2+1\,\overline{f}_3=0$. Having calculated such relationships in all dimensions up to 40, in all even dimensions there is a single relationship, in all odd dimensions there is a two-dimensional family of relationships. We come upon connections to Tchebyshev, Bernoulli, \& Euler polynomials, and Stirling numbers.