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### Problem 57: \( f(x) = 2x^3 + 3x^2 - 8x + 5 \)
We need to find the possible rational zeros using the **Rational Root Theorem**.

1. **Coefficients**:
   - Leading coefficient: \( a_0 = 2 \)
   - Constant term: \( a_n = 5 \)

2. **Possible rational roots**:
   Possible rational roots are given by the formula \( \frac{p}{q} \), where:
   - \( p \) is a divisor of the constant term \( a_n = 5 \)
   - \( q \) is a divisor of the leading coefficient \( a_0 = 2 \)

   Divisors of 5 (constant term \( a_n \)): \( \pm 1, \pm 5 \)

   Divisors of 2 (leading coefficient \( a_0 \)): \( \pm 1, \pm 2 \)

   Therefore, the possible rational roots are:

   \[
   \pm 1, \pm \frac{1}{2}, \pm 5, \pm \frac{5}{2}
   \]

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### Problem 59: \( f(x) = 6x^4 - 10x^2 + 13x + 1 \)
Again, we apply the **Rational Root Theorem**.

1. **Coefficients**:
   - Leading coefficient: \( a_0 = 6 \)
   - Constant term: \( a_n = 1 \)

2. **Possible rational roots**:
   Using the Rational Root Theorem, the possible roots are \( \frac{p}{q} \), where:
   - \( p \) is a divisor of the constant term \( a_n = 1 \)
   - \( q \) is a divisor of the leading coefficient \( a_0 = 6 \)

   Divisors of 1 (constant term \( a_n \)): \( \pm 1 \)

   Divisors of 6 (leading coefficient \( a_0 \)): \( \pm 1, \pm 2, \pm 3, \pm 6 \)

   Therefore, the possible rational roots are:

   \[
   \pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}
   \]

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Would you like further details on how to test these roots or any other part of the process? Here are some questions that build on this information:

1. How do you use synthetic division to test these possible rational roots?
2. Can the Rational Root Theorem guarantee all the zeros of the polynomial?
3. How does the degree of the polynomial affect the number of possible real roots?
4. What are the differences between rational and irrational roots in polynomial equations?
5. How would you factor the polynomial once a rational root is found?

**Tip**: Testing rational roots is usually done by substituting each value into the function until one gives zero, indicating it's a root.

Answer 57 and 59 from the given polynomial equations

Find the possible rational roots for the polynomials f(x) = 2x^3 + 3x^2 - 8x + 5 and f(x) = 6x^4 - 10x^2 + 13x + 1 using the Rational Root Theorem. Explore algebraic concepts including leading coefficients and constant terms. Suitable for high school students in Grades 10-12.

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