Let $$x^3 + bx^2 + cx + d$$ be a polynomial with coefficients in $\mathbf{Q}$. We ask which condition(s) $b,c,d$ must satisfy in order that one ( any ) root be the average of the other two roots?
Solution.
Let the roots of $x^3 + bx^2 + cx + d = 0$ be $\alpha_1, \alpha_2, \alpha_3$. Clearly, the roots must satisfy the following equations:
\begin{align*}
\alpha_1=& \frac{1}{2}\alpha_2 + \frac{1}{2}\alpha_3 \\
\alpha_2=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_3 \\
\alpha_3=& \frac{1}{2}\alpha_1 + \frac{1}{2}\alpha_2 \\
\end{align*}
This is a rank $2$ system of linear equations with solution $( \alpha_1, \alpha_2, \alpha_3) = \lambda (1,1,1) $, and implies that the equation has three equal roots and can thus be written as follows:$$(x-\alpha)^3 = x^3-3\alpha x^2 + 3\alpha^2 x - \alpha^3$$.
So if the root is $\alpha$ then $b=-3\alpha, c=3\alpha^2$ and $d=-\alpha^3$.
