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The problem asks us to use the **Rational Zeros Theorem** to list all possible zeros for the polynomial function:

\[
f(x) = 5x^3 - 2x^2 + 5x + 7
\]

### Rational Zeros Theorem:
The Rational Zeros Theorem states that the possible rational zeros of a polynomial are given by:

\[
\text{Possible zeros} = \pm \frac{\text{factors of the constant term}}{\text{factors of the leading coefficient}}
\]

- The **constant term** in this polynomial is \( 7 \).
- The **leading coefficient** is \( 5 \) (the coefficient of \( x^3 \)).

### Step 1: Find factors of the constant term (7)
The factors of 7 are: \( \pm 1, \pm 7 \).

### Step 2: Find factors of the leading coefficient (5)
The factors of 5 are: \( \pm 1, \pm 5 \).

### Step 3: Form all possible combinations
Using the Rational Zeros Theorem, the possible rational zeros are:

\[
\pm \frac{1}{1}, \pm \frac{7}{1}, \pm \frac{1}{5}, \pm \frac{7}{5}
\]

Which simplifies to:

\[
\pm 1, \pm 7, \pm \frac{1}{5}, \pm \frac{7}{5}
\]

### Final List of Possible Rational Zeros:
The possible zeros of the function are:

\[
1, -1, 7, -7, \frac{1}{5}, -\frac{1}{5}, \frac{7}{5}, -\frac{7}{5}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you verify if a given possible zero is an actual zero?
2. What are irrational zeros, and how do they differ from rational zeros?
3. What role do complex numbers play in finding the zeros of a polynomial?
4. Can the Rational Zeros Theorem be applied to higher-degree polynomials (degree 4 and above)?
5. What other methods can be used to find zeros of a polynomial apart from the Rational Zeros Theorem?

### Tip:
Once you have the list of possible rational zeros, you can use synthetic division or substitution to verify which of them are actual zeros.

Use the rational zeros theorem to list all possible zeros of the function f(x) = 5x^3 - 2x^2 + 5x + 7. Enter the possible zeros separated by commas. You do not need to factor the polynomial.

Explore how to apply the Rational Zeros Theorem to list all possible zeros of the cubic polynomial f(x) = 5x^3 - 2x^2 + 5x + 7. This problem covers key algebraic concepts and demonstrates how to find rational solutions for polynomials. Ideal for high school students in Grades 9-12.

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