Verify that the negative root of x³-2x²-5x+6 = 0 lies in the interval [-3,-1] using the newton - Raphson method

To verify that the negative root of \(x^3 - 2x^2 - 5x + 6 = 0\) lies in the interval \([-3, -1]\) using the Newton-Raphson method, we can follow these steps:

1. **Define the function and its derivative:**
   \[
   f(x) = x^3 - 2x^2 - 5x + 6
   \]
   \[
   f'(x) = 3x^2 - 4x - 5
   \]

2. **Check the signs of the function at the endpoints of the interval \([-3, -1]\):**
   \[
   f(-3) = (-3)^3 - 2(-3)^2 - 5(-3) + 6 = -27 - 18 + 15 + 6 = -24
   \]
   \[
   f(-1) = (-1)^3 - 2(-1)^2 - 5(-1) + 6 = -1 - 2 + 5 + 6 = 8
   \]

   Since \( f(-3) < 0 \) and \( f(-1) > 0 \), there is a sign change between \(-3\) and \(-1\), indicating that there is at least one root in the interval \([-3, -1]\).

3. **Apply the Newton-Raphson method with an initial guess within the interval \([-3, -1]\):**

   Let's start with \(x_0 = -2\) (since it's a midpoint within \([-3, -1]\)).

4. **Perform Newton-Raphson iterations:**
   \[
   x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}
   \]

   - **Iteration 1:**
     \[
     x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = -2 - \frac{(-2)^3 - 2(-2)^2 - 5(-2) + 6}{3(-2)^2 - 4(-2) - 5}
     \]
     \[
     f(-2) = -8 - 8 + 10 + 6 = 0
     \]
     \[
     f'(-2) = 3(4) + 8 - 5 = 15
     \]
     \[
     x_1 = -2 - \frac{0}{15} = -2
     \]

   In this case, \(x_0 = -2\) is already a root because \( f(-2) = 0 \).

5. **Conclusion:**
   Since the Newton-Raphson method converges to \(x = -2\) starting from \(x_0 = -2\), we can confirm that there is a negative root at \(x = -2\), which lies within the interval \([-3, -1]\).

Would you like more details or have any questions about this verification process?

### Additional Questions:
1. How does the Newton-Raphson method ensure that a root is within a given interval?
2. What other numerical methods can be used to verify the existence of roots within an interval?
3. How does the choice of the initial guess affect the convergence of the Newton-Raphson method?
4. Can the Newton-Raphson method fail to find a root even if it exists within the interval?
5. What are the benefits of using the Newton-Raphson method over other root-finding methods?
6. How can we modify the Newton-Raphson method to ensure convergence for functions with multiple roots?
7. What are some practical applications of the Newton-Raphson method in real-world problems?
8. How do we handle cases where the derivative of the function is zero at the initial guess?

### Tip:
When using the Newton-Raphson method, always graph the function and its derivative to gain insight into the behavior of the function and choose a suitable initial guess. This helps in ensuring the method converges to the desired root.

Learn how to verify the existence of the negative root of the polynomial equation x³ - 2x² - 5x + 6 = 0 within the interval [-3, -1] using the Newton-Raphson method. Understand the steps involved, including function and derivative definition, initial guess selection, and convergence analysis.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation