Polynomials

I am studying some more about polynomials, the topic of symmetric polynomials for example, is an interesting one.

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?

To be continued.