If c is a zero of a polynomial, then by the remainder theorem, x-c is a factor of the polynomial. Since our polynomial has zeros at -7, 4, and 6, we can write it as (x+7)(x-4)(x-6). Foiling the first two factors, we get (x^2 + 3x - 28)(x - 6). Distributing again, we get X^3 + 3x^2 - 28x - 6x^2 - 18x + 168. Finally, combining like terms gives us:
x^3 - 3x^2 - 46x + 168, which has degree 3 and a coefficient of 1 on X^3, so we are done.