To solve this problem, we need to apply the Rational Zeros Theorem to the polynomial function:

$f(x) = 2x^7 - x^5 + 5x^4 + 28$

Step-by-Step Explanation

1. Identify the coefficients:

   • The leading coefficient is $2$ (coefficient of $x^7$).
   • The constant term is $28$.

2. Rational Zeros Theorem:

   • The potential rational zeros are in the form $\frac{p}{q}$, where:
     • $p$ is a factor of the constant term (28).
     • $q$ is a factor of the leading coefficient (2).

3. Factors of the constant term (28): $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28$

4. Factors of the leading coefficient (2): $\pm 1, \pm 2$

5. Form all combinations $\frac{p}{q}$: $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{14}{2}, \pm \frac{28}{2}$ Simplifying the fractions: $\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm \frac{1}{2}, \pm \frac{7}{2}$

Answer Verification

From the options provided:

• Option D matches the correct potential rational zeros list, including $\pm \frac{1}{28}$.

Correct Answer: D

Would you like a detailed explanation on any part of the solution?

Related Questions:

1. How do you find factors of a number for the Rational Zeros Theorem?
2. What happens if the polynomial has a leading coefficient of 1?
3. Can the Rational Zeros Theorem identify all real zeros of a polynomial?
4. What is the next step if you want to verify whether a potential zero is an actual zero?
5. How do you perform synthetic division to test potential zeros?

Tip:

Always check the simplification of fractions when listing potential rational zeros to avoid redundancy.