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The question asks for the approximate value of one root of the polynomial equation:
\[
x^3 - 2x^2 - 5x + 6 = 0
\]
using two methods:

1. **Bisection Method** with initial interval \( a = 0.8 \) and \( b = 1.4 \).
2. **Newton-Raphson Method** with initial guess \( x_0 = 1.5 \).

The approximation process should continue until the True Percent Relative Error \( | \epsilon_t | \) and Approximate Percent Relative Error \( | \epsilon_a | \) are both below \( \epsilon_s = 0.05\% \).

### Steps:
1. **For the Bisection Method**:
   - Start with the interval \([0.8, 1.4]\) and repeatedly apply the bisection formula:
     \[
     x_{\text{mid}} = \frac{a + b}{2}
     \]
   - Calculate \( f(a) \), \( f(b) \), and \( f(x_{\text{mid}}) \).
   - Narrow down the interval based on the sign of \( f(x_{\text{mid}}) \), then update \( a \) or \( b \).
   - Continue until \( | \epsilon_a | < 0.05\% \).

2. **For the Newton-Raphson Method**:
   - Begin with \( x_0 = 1.5 \) and apply the iterative formula:
     \[
     x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
     \]
   - Calculate the derivative \( f'(x) = 3x^2 - 4x - 5 \).
   - Continue updating \( x_n \) until \( | \epsilon_a | < 0.05\% \).

Would you like a detailed solution for each method? 

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### Additional Questions to Explore:

1. What are the benefits of using the Bisection Method over the Newton-Raphson Method, and vice versa?
2. How does the choice of initial interval affect the convergence in the Bisection Method?
3. Why is the Newton-Raphson method sensitive to the initial guess?
4. What are other numerical methods to find roots of polynomials?
5. How can we ensure the method chosen will converge to the desired root?

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### Tip
For the Newton-Raphson Method, ensure the function's derivative does not approach zero near the initial guess, as this can cause divergence or slow convergence.

Tentukan nilai aproksimasi salah satu akar dari: x^3 - 2x^2 - 5x + 6 = 0 menggunakan: a) Metode bisection dengan dua titik awalnya a = 0,8 dan 1,4 b) Metode Newton-Raphson dengan satu titik awal x0 = 1,5 Hitung sampai nilai aproksimasinya sampai True percent relative error |et| < es dan Approximate percent relative error |ea| < es, dimana es = 0,05%

Learn how to approximate one root of the polynomial equation x^3 - 2x^2 - 5x + 6 using numerical methods. The solution covers the Bisection Method with the initial interval of [0.8, 1.4] and the Newton-Raphson Method with the initial guess x0 = 1.5, continuing until the true percent relative error and approximate percent relative error are below 0.05%.

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