Let $p(x) = x^2 + ax + b$ have two distinct real roots, where $a,b$ are real numbers. Define $g(x) = p(x^3)$ for all real $x$. Then, which of the following statements are true?

I. $g$ has exactly two distinct real roots.
II. $g$ can have more than two distinct real roots.
III. There exists a real number $\alpha$ such that $g(x) \ge \alpha$ for all real $x$.