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The problem in the image is asking to list all possible rational zeros of the polynomial function \( f(x) = 10x^5 - 17x^4 - 8x^3 - 4x^2 + 6x - 1 \) using the **Rational Zero Theorem**. Let's walk through the steps of this theorem to find the answer.

### Rational Zero Theorem
The Rational Zero Theorem states that any rational zero of a polynomial will be of the form:

\[
\frac{p}{q}
\]

where:
- \( p \) is a factor of the constant term (the last term, in this case, \(-1\)).
- \( q \) is a factor of the leading coefficient (the coefficient of the highest degree term, in this case, \(10\)).

### Step 1: Find factors of the constant term \(-1\)
The factors of \(-1\) are:
\[
p = \pm 1
\]

### Step 2: Find factors of the leading coefficient \(10\)
The factors of \(10\) are:
\[
q = \pm 1, \pm 2, \pm 5, \pm 10
\]

### Step 3: List all possible rational zeros
The possible rational zeros are all combinations of \( \frac{p}{q} \), where \( p = \pm 1 \) and \( q = \pm 1, \pm 2, \pm 5, \pm 10 \). This gives:

\[
\text{Possible rational zeros} = \pm 1, \pm \frac{1}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}
\]

### Step 4: Compare the options
Based on the possible rational zeros, the correct answer should list:

\[
\{-1, 1, -\frac{1}{2}, \frac{1}{2}, -\frac{1}{5}, \frac{1}{5}, -\frac{1}{10}, \frac{1}{10}\}
\]

Now, looking at the options in the image:
- **Option C** matches this list.

### Final Answer:
The correct answer is **C**.

---

Would you like further details or have any other questions?

Here are 5 related questions you might find helpful:
1. How can we verify if one of these possible zeros is actually a zero of the polynomial?
2. What is the next step after listing possible rational zeros?
3. How does synthetic division help in finding zeros of a polynomial?
4. Can a polynomial have non-rational zeros, and how are they found?
5. What happens if all rational zeros are tested and none work?

**Tip:** Always test the possible rational zeros by substituting them back into the polynomial to see if they satisfy \( f(x) = 0 \).

Use the Rational Zero Theorem to list all possible rational zeros for the given function: f(x) = 10x^5 - 17x^4 - 8x^3 - 4x^2 + 6x - 1.

Learn how to list all possible rational zeros for the polynomial f(x) = 10x^5 - 17x^4 - 8x^3 - 4x^2 + 6x - 1 using the Rational Zero Theorem. This problem covers polynomial concepts and the application of the Rational Zero Theorem, suitable for high school math students.

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