To use the Rational Zeros Theorem to list all possible zeros for the polynomial $f(x) = 2x^3 - 5x^2 - 2x + 5$, we proceed as follows:

1. Identify the leading coefficient and constant term:

   • The leading coefficient (coefficient of the highest degree term) is $2$.
   • The constant term is $5$.

2. Possible rational zeros: According to the Rational Zeros Theorem, any possible rational zero is of the form $\pm \frac{p}{q}$, where:

   • $p$ is a factor of the constant term (5).
   • $q$ is a factor of the leading coefficient (2).

3. Factors of the constant term (5):

   • The factors of 5 are $\pm 1$ and $\pm 5$.

4. Factors of the leading coefficient (2):

   • The factors of 2 are $\pm 1$ and $\pm 2$.

5. List all possible values of $\frac{p}{q}$:

   • Taking each combination of factors of 5 over factors of 2, we get: $\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}$

Therefore, the possible rational zeros are: $\pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}$

You can enter these values in the answer box as: 1, -1, 1/2, -1/2, 5, -5, 5/2, -5/2

Would you like more details on this solution, or do you have any other questions?

Here are some related questions you might find helpful:

1. How do we test which of these possible zeros are actual zeros?
2. What if none of the rational zeros work? What methods can we use next?
3. Can the Rational Zeros Theorem be applied to polynomials with irrational or complex zeros?
4. How does synthetic division help verify which values are actual zeros?
5. Why is it useful to list all possible rational zeros even if some may not be actual zeros?

Tip: When listing possible rational zeros, always simplify fractions and remember to include both positive and negative values.