To determine the best list of possible zeros for the function $f(x) = x^3 + x^2 - 13x - 6$, we can apply the Rational Root Theorem. This theorem states that any rational zero of a polynomial, $\frac{p}{q}$, must be a factor of the constant term divided by a factor of the leading coefficient.

Step-by-Step Solution:

1. Identify the Leading Coefficient and Constant Term:

   • The leading coefficient (coefficient of $x^3$) is $1$.
   • The constant term is $-6$.

2. Factors of the Constant Term (-6):

   • The factors of $-6$ are $\pm 1, \pm 2, \pm 3, \pm 6$.

3. Factors of the Leading Coefficient (1):

   • The factors of $1$ are $\pm 1$.

4. Possible Rational Zeros:

   • By the Rational Root Theorem, the possible rational zeros are $\frac{\text{factors of } -6}{\text{factors of } 1}$.
   • This gives us the possible zeros: $\pm 1, \pm 2, \pm 3, \pm 6$.

5. Match with the Answer Choices:

   • The correct list of possible zeros is therefore $\pm 1, \pm 2, \pm 3, \pm 6$, which corresponds to option (d).

Final Answer:

Option (d): $\pm 1, \pm 2, \pm 3, \pm 6$.

Would you like a more detailed explanation of any step?

Here are some related questions to explore:

1. How do you determine the possible rational roots for other polynomial functions?
2. Can irrational or complex roots appear in this function? How can you tell?
3. What are synthetic division and the factor theorem, and how can they help in finding actual roots?
4. How does the Rational Root Theorem work with polynomials that have higher leading coefficients?
5. What is the relationship between zeros of a function and its graph?

Tip: The Rational Root Theorem is useful for quickly narrowing down potential roots before attempting factorization or synthetic division.