The problem is asking to list all possible rational zeros for the function $f(x) = x^3 - 11x^2 - 14x + 14$, using the Rational Zero Theorem.

Rational Zero Theorem

This theorem states that any rational zero of a polynomial function is of the form: $\frac{p}{q}$ where:

• $p$ is a factor of the constant term (here $14$),
• $q$ is a factor of the leading coefficient (here $1$).

Step 1: Identify factors of the constant and leading coefficient

• Constant term: $14$, factors: $\pm 1, \pm 2, \pm 7, \pm 14$
• Leading coefficient: $1$, factors: $\pm 1$

Step 2: Form the possible rational zeros

The possible rational zeros are: $\pm \frac{1}{1}, \pm \frac{2}{1}, \pm \frac{7}{1}, \pm \frac{14}{1}$ This simplifies to: $\pm 1, \pm 2, \pm 7, \pm 14$

Step 3: Compare with options

From the options provided, option C correctly lists the possible rational zeros: $-1, 1, -2, 2, -7, 7, -14, 14$.

Thus, the correct answer is C.

Would you like more details or explanations?

Here are 5 related questions:

1. How does the Rational Zero Theorem help find rational zeros?
2. What are the conditions for applying the Rational Zero Theorem?
3. Why are the signs both positive and negative for possible zeros?
4. What happens if no rational zeros are found from the list?
5. How can synthetic division help confirm possible zeros?

Tip: When using the Rational Zero Theorem, always list factors of both the constant and leading coefficient to form all possible rational zeros.