Answer to POW #2:

Let a be the number in question. By cubing a and simplifying, we obtain a^3 = 14 - 3a. Setting equal to zero, guessing the root of a=2, and using synthetic division, we obtain (a - 2)(a^2 + 2a + 7) = 0. The quadratic part has no real solutions (since 2^2 - 4(1)(7) < 0), and thus a=2 is the only real root of the equation. Therefore the number in question (a) is an integer (in fact, it is 2).

Source: From the Purdue Math Department. (And, as pointed out to me by Carl Droms, this type of number arises in Cardano's solution to the cubic equation.)