Let x^3 + bx^2 + cx + dbe 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.
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