Question #40878

Let $O$ be the origin. The coordinates of three points $A, B,$ and $C$ are determined by the roots of the following equations: For $A$: $t^3 - 7t + 6 = 0$ For $B$: $t^3 + 2t^2 - 5t - 6 = 0$ For $C$: $t^3 + 3t^2 - 4t - 12 = 0$ The coordinates $(x, y, z)$ of each point are a permutation of the roots of their respective equations, satisfying the following conditions: (i) $A$ lies in Octant II, and its $y$-coordinate is strictly less than its $z$-coordinate. (ii) $B$ lies in Octant VIII, and its $y$-coordinate is strictly greater than its $z$-coordinate. (iii) $C$ lies in Octant VI, and the absolute value of its $x$-coordinate is strictly greater than the absolute value of its $z$-coordinate. Let $V$ be the volume of the parallelepiped with coterminous edges $\vec{OA}, \vec{OB},$ and $\vec{OC}$. If $S = [\vec{OA}\ \vec{OB}\ \vec{OC}]$ is their scalar triple product, find the value of $2V - S$.