Question 37
A cubic trinomial $x^3 + px + q$ with integer coefficients $p$ and $q$ is said to be *irrational* if it has three pairwise distinct real irrational roots $\alpha_1$, $\alpha_2$, $\alpha_3$.
Find all irrational cubic trinomials for which the value of $|\alpha_1| + |\alpha_2| + |\alpha_3|$ is the minimal possible.
Find all irrational cubic trinomials for which the value of $|\alpha_1| + |\alpha_2| + |\alpha_3|$ is the minimal possible.