To determine if the statement is true or false, we need to verify whether the given list includes all possible rational roots for the function $f(x) = 3x^4 + x^3 - 6x^2 - 14x - 12$.

Step 1: Apply the Rational Root Theorem

The Rational Root Theorem states that any possible rational root of a polynomial with integer coefficients is of the form: $\frac{p}{q}$ where:

• $p$ is a factor of the constant term (-12).
• $q$ is a factor of the leading coefficient (3).

Step 2: Identify the Factors of the Constant and Leading Coefficient

• Factors of $-12$ are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.
• Factors of $3$ are $\pm 1, \pm 3$.

Step 3: Determine Possible Rational Roots

By taking each combination of $\frac{p}{q}$, the possible rational roots are: $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}$

Step 4: Compare with the Given List

The provided list in the question is: $\pm 1, \pm 2, \pm 4, \pm 12, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}$ This list excludes $\pm 3$ and $\pm 6$, which are also possible rational roots according to the Rational Root Theorem.

Conclusion

Since the list provided does not include all possible rational roots, the correct answer is: False

Would you like more details on any part of this process, or do you have additional questions?

Here are 5 related questions you might find useful:

1. How do we determine if a polynomial has rational roots?
2. What other methods can we use to solve for roots of a polynomial?
3. Can a polynomial have irrational or complex roots in addition to rational ones?
4. What is the importance of the Rational Root Theorem in polynomial factorization?
5. How would the solution change if the leading coefficient was different?

Tip: Always use the Rational Root Theorem first when checking for possible rational roots—it can save time in identifying all possibilities quickly.