Centroid of the roots

Problem

Knowing that the roots of $x^3+px+q=0$ sum to $0$ (because $a_2=0$), show that the centroid of the three roots in the Argand-Gauss plane is always the origin. (It holds for every depressed polynomial!)

Solution

In the polynomial $x^3 + 0\cdot x^2 + px + q$ the degree-$2$ term is missing, so $a_2=0$ and by Vieta $\sigma_1 = r_1+r_2+r_3 = -\frac{a_2}{a_3} = 0.$ The centroid of the three points representing the roots in the Argand-Gauss plane is their arithmetic mean: $G = \frac{r_1+r_2+r_3}{3} = \frac{0}{3} = \boxed{0}.$ It therefore coincides with the origin. The same reasoning holds for every depressed polynomial (without the degree-$n-1$ term): if $a_{n-1}=0$ then $\sigma_1=0$ and the centroid of the $n$ roots is the origin.

Links

Topics: Complex numbers
Concepts: Centroid · Argand-Gauss plane · Vieta’s relations
Skills: Proving
Exercise type: Proof