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The question requires using the **Rational Zero Theorem** to list all possible rational zeros for the given polynomial function:

\[
f(x) = 5x^4 + 9x^3 - 20x^2 + 15x - 10
\]

### **Steps to Solve:**

1. **Identify the leading coefficient and constant term**:
   - Leading coefficient (of \(x^4\)) = **5**
   - Constant term = **-10**

2. **Rational Zero Theorem**:
   The rational zeros are of the form:
   \[
   \frac{p}{q}
   \]
   where:
   - \(p\) is a factor of the constant term (-10)
   - \(q\) is a factor of the leading coefficient (5)

3. **Find factors**:
   - Factors of **-10** (constant term): \(\pm 1, \pm 2, \pm 5, \pm 10\)
   - Factors of **5** (leading coefficient): \(\pm 1, \pm 5\)

4. **List all possible rational zeros**:
   All possible rational zeros are:
   \[
   \pm \frac{1}{1}, \pm \frac{1}{5}, \pm \frac{2}{1}, \pm \frac{2}{5}, \pm \frac{5}{1}, \pm \frac{5}{5}, \pm \frac{10}{1}, \pm \frac{10}{5}
   \]
   Simplifying, this becomes:
   \[
   \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5}
   \]

### **Conclusion**:
The correct answer is **option A**, which includes:
\[
\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{5}, \pm \frac{2}{5}
\]

Would you like further details or have any questions?

### **Related Questions**:
1. What are the next steps after finding the list of possible rational zeros?
2. How do you check which of these possible zeros are actual roots of the polynomial?
3. What role does synthetic division play in verifying rational zeros?
4. Can irrational or complex roots be part of the solution, even if the Rational Zero Theorem lists rational ones?
5. How does the Rational Zero Theorem change for polynomials with higher degrees or different coefficients?

**Tip**: When using the Rational Zero Theorem, always remember to simplify the fractions formed by \(\frac{p}{q}\).

Use the Rational Zero Theorem to list all possible rational zeros for the given function: f(x) = 5x^4 + 9x^3 - 20x^2 + 15x - 10

Learn how to use the Rational Zero Theorem to find all possible rational zeros for the polynomial function f(x) = 5x^4 + 9x^3 - 20x^2 + 15x - 10. This solution covers algebraic techniques and factorization rules for high school math students.

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