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 $f(x) = 0$.