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Background: It's straightforward to check that the average (i.e. the mean) of the roots of a nonlinear polynomial equals the average of the roots of its derivative: if

$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots a_0$$

then the roots of $f(x)$, counting multiplicities, sum to $- a_{n-1}$, while the roots of

$$f'(x) = n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1$$

sum to

$$\frac{- (n-1) a_{n-1}}{n},$$

so that both averages are equal to $- a_{n-1}/n,$ and multiplying through to get a non-monic polynomial obviously doesn't affect this.

Question: What is the earliest known reference to this particular fact in writing? (The earliest I've found have only been 10 or 20 years old, so even if something isn't known to be the oldest it's helpful so long as it's older than anything posted here.)

Bonus question: Are there references to the fact that the $(n-1)$-coefficient of a monic polynomial is the sum of its roots significantly predating Viete's characterization of the coefficients of a polynomial in terms of its roots?

Motivation: a calculus student who rediscovered and proved this fact asked in the comments here:

http://www.reddit.com/r/math/comments/1uqaiv/getting_credit_for_original_work/

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For $f$, the zeroes sum up to $(-1)^{n-1}a_{n-1}$, don't they? –  Michael Hoppe Jan 9 at 6:56
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@MichaelHoppe Nope. –  user119256 Jan 9 at 6:58
    
@Karene Your'e right, I haven't had my morning coffee yet. –  Michael Hoppe Jan 9 at 7:00
    
Observation: the claim is not true if one only considers real roots: $f(x)=x^3+1$. –  Michael Hoppe Jan 9 at 7:26
    
As a consequence, the inflection point's first coordinate of a function of degree three is the average of it's zeroes. Didn't noticed that before. –  Michael Hoppe Jan 9 at 7:43

1 Answer 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.

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