I have a short note recently accepted to American Math. Monthly but it is all too easy and surely is a rediscovery. The result is that the Lines of best fit for the zeros and for the critical points of a univariate polynomial coincide. Proof at http://school.maths.uwa.edu.au/~keady/Torsion/MomInertia/MAA10/ammKeady10.pdf

QUESTION. Has the result been published before? If so, where.

I regard the paper as being about "second-moments" ... and people are asking me what is known/might be true for "third moments" (and except for a bit of Newton Identity stuff, I don't know).

For me this is "hobbyist maths". To earn my living I really am meant to do more serious "applied maths". "Hobbyist maths" it might be: I still want to know more about it all. AsI have yet to see "page proofs"/etc. from Amer. Math. Monthly, I could get references on prior publication into it, if Im given them soon.