Use the rational zeros theorem to list all possible zeros of the function.       f(x)=3x^3-3x^2-4x+5  .      Enter the possible zeros separated by commas. You do not need to factor the polynomial.

To use the **Rational Zeros Theorem**, we need to find the possible rational zeros of the polynomial function:

\[ f(x) = 3x^3 - 3x^2 - 4x + 5 \]

### Steps to find the possible zeros:
1. **Identify the constant term (p)**: This is the last term of the polynomial, which is \( 5 \).
2. **Identify the leading coefficient (q)**: This is the coefficient of the highest degree term, which is \( 3 \).

### The Rational Zeros Theorem tells us that the possible rational zeros are the factors of the constant term \( p \) divided by the factors of the leading coefficient \( q \). That is:

\[
\text{Possible zeros} = \frac{\text{Factors of } p}{\text{Factors of } q}
\]

#### Factors of \( p = 5 \): \( \pm 1, \pm 5 \)

#### Factors of \( q = 3 \): \( \pm 1, \pm 3 \)

Now, we list all the possible combinations of \( \frac{p}{q} \):

\[
\text{Possible rational zeros} = \pm 1, \pm \frac{1}{3}, \pm 5, \pm \frac{5}{3}
\]

### Conclusion:
The possible rational zeros of the function \( f(x) = 3x^3 - 3x^2 - 4x + 5 \) are:
\[
1, -1, \frac{1}{3}, -\frac{1}{3}, 5, -5, \frac{5}{3}, -\frac{5}{3}
\]

If you have any questions or want to go deeper into this, feel free to ask! 

Here are 5 related questions to help you understand more:
1. What is the Rational Zeros Theorem, and why is it useful?
2. How can you verify which of the possible zeros are actual zeros?
3. What is the process for dividing polynomials to check potential zeros?
4. Can a polynomial have no rational zeros? Why?
5. How would the Rational Zeros Theorem change if the polynomial had a different leading coefficient?

**Tip**: Always simplify your factors as much as possible before listing potential rational zeros!

Use the rational zeros theorem to list all possible zeros of the function. f(x)=3x^3-3x^2-4x+5. Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Learn how to use the Rational Zeros Theorem to find the possible rational zeros of the cubic polynomial f(x) = 3x^3 - 3x^2 - 4x + 5. This problem covers key concepts in polynomial functions and factorization, suitable for students in Grades 10-12.

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