10.11. What condition must the coefficients of the polynomial $x^{3}+a x^{2}+b x+c$ satisfy so that its three roots form an arithmetic progression?
10.11. The numbers $x_{1}, x_{2}, x_{3}$ form an arithmetic progression if and only if $x_{1}+x_{3}=2 x_{2}$ (possibly after renumbering), i.e., $x_{1}+x_{2}+x_{3}=3 x_{2}$. According to Vieta's theorem, $x_{1}+x_{2}+x_{3}=-a$. Therefore, the number $x_{2}=-a / 3$ must be a root of the given polynomial. An equivalent condition is: $\frac{2}{27} a^{3}-\frac{a b}{3}+c=0$.