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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.

Use the rational zeros theorem to list the potential rational zeros of the polynomial function. Do not attempt to find the zeros.

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

Find the potential rational zeros. Choose the correct answer below:

A. -1, 1, -2, 2, -4, 4, -7, 7, -14, 14, -28, 28, -1/2, 1/2, -1/28, 1/28

B. -1, 1, -2, 2, -4, 4, -7, 7, -14, 14, -28, 28, -1/2, 1/2, -7/2, 7/2

C. -1, 1, -2, 2, -4, 4, -7, 7, -28, 28, -1/2, 1/2, -1/28, 1/28

D. -1, 1, -2, 2, -4, 4, -7, 7, -28, 28, -1/2, 1/2, -1/28, 1/28

Learn how to apply the Rational Zeros Theorem to list all possible rational zeros of the polynomial function f(x) = 2x^7 - x^5 + 5x^4 + 28. This solution provides a step-by-step method for finding the potential rational zeros using factors of the constant term and leading coefficient. Suitable for high school algebra students.

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