For this exercise, consider the polynomial shown below: Note Make sure multiple answers are separated by commas. The factored form is a single answer and therefore does not require commas.
\[
f(x)=x^{3}-3 x^{2}-4 x+12
\]
List ALL possible rational roots for $f(x)$ :
\[
\frac{p}{q}=
\]
Answers should be integers or exact fractions. Do not use decimals!
Now list the ACTUAL roots for $f(x)$ :
\[
x=
\]
Answers should be integers or exact fractions. Do not use decimals!
The fully factored form of the polynomial is:
\[
f(x)=
\]

Step 1 ：Find the factors of the constant term (12) and the leading coefficient (1): \(p_{\text{factors}} = \{1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12\}\), \(q_{\text{factors}} = \{1, -1\}\)

Step 2 ：Form all possible combinations of p/q and simplify: \(\boxed{1, 2, 3, 4, 6, 12, -1, -2, -3, -4, -6, -12}\)

Step 3 ：Now, we need to find the actual roots of the polynomial. We can do this by substituting each possible root into the polynomial and checking if it equals zero. After checking, we find that the actual roots are \(\boxed{3, -2, 2}\)

Step 4 ：The fully factored form of the polynomial is: \(f(x) = (x-3)(x+2)(x-2)\)