x^{3}-3 x^{2}-4 x+12=0 Use synthetic division to test several possible rational roots in order to identify one actual root

Step 1 ：Given the equation \(x^{3}-3 x^{2}-4 x+12=0\)

Step 2 ：According to the rational root theorem, the possible rational roots of a polynomial equation are given by the factors of the constant term divided by the factors of the leading coefficient. In this case, the constant term is 12 and the leading coefficient is 1. Therefore, the possible rational roots are ±1, ±2, ±3, ±4, ±6, and ±12.

Step 3 ：We can use synthetic division to test these possible roots. If the remainder is zero, then we have found a root.

Step 4 ：After testing, we find that the actual roots of the equation are 2, -2, and 3.

Step 5 ：Final Answer: The actual roots of the equation \(x^{3}-3 x^{2}-4 x+12=0\) are \(\boxed{2}\), \(\boxed{-2}\), and \(\boxed{3}\).