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The question asks: "Which of the sets below shows the list of possible rational zeroes of \( 2x^3 - 3x^2 + 4x + 5 \)?"

To find the list of possible rational zeroes, we apply the **Rational Root Theorem**. The Rational Root Theorem suggests that any possible rational root is of the form:

\[
\frac{p}{q}
\]
where:
- \( p \) is a factor of the constant term (in this case, 5).
- \( q \) is a factor of the leading coefficient (in this case, 2).

### Step 1: Find factors of the constant term (5)
The factors of 5 are:
\[
p = \pm 1, \pm 5
\]

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

### Step 3: List possible rational roots
The possible rational roots are:
\[
\frac{p}{q} = \pm 1, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2}
\]

So, the set of possible rational zeroes is:
\[
\left\{ \pm 1, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2} \right\}
\]

Would you like more details on the Rational Root Theorem or the next steps in solving the equation?

Here are five questions related to this topic:
1. How do you determine if a possible rational root is an actual root of the polynomial?
2. Can the Rational Root Theorem be used for polynomials with irrational or complex roots?
3. What other methods can be used to find the roots of a cubic polynomial?
4. How do synthetic division and the Rational Root Theorem work together to factor polynomials?
5. What does it mean if none of the possible rational roots satisfy the equation?

**Tip**: Always verify the rational roots by substituting them into the polynomial to check if they make the equation equal zero.

Which of the sets below shows the list of possible rational zeroes of 2x^3 - 3x^2 + 4x + 5?

Learn how to find the list of possible rational zeroes for the cubic polynomial 2x^3 - 3x^2 + 4x + 5 using the Rational Root Theorem. Explore step-by-step solutions suitable for high school students, focusing on algebraic concepts and polynomial root-finding techniques.

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