Use Newton's method to find all the roots of the equation correct to eight decimal places. Start by drawing a graph to find initial approximations. x^6 - x^5 - 6x^4 - x^2 + x + 8 = 0
Hint:
There are 4 real roots and 2 complex roots. The 4 real roots lie between -2 and +3.
Also, read:
http://www.math.montana.edu/frankw/ccp/calculus/numerical/newton/learn.htm
as suggested by Mr. Pursley.
See also example:
http://www.jiskha.com/display.cgi?id=1288905148
To find the roots of the equation using Newton's method, we will first start by drawing a graph to find initial approximations.
1. Graph the equation:
Plot the function f(x) = x^6 - x^5 - 6x^4 - x^2 + x + 8 on a graphing tool or software.
Alternatively, you can sketch a rough graph by plugging in some values of x and observing the behavior of the function.
2. Locate the approximate x-values where the function crosses the x-axis. These crossings correspond to the roots of the equation.
3. Choose initial approximations:
For each root, choose an initial approximation by looking at the x-values where the function crosses the x-axis.
Choose reasonably accurate initial approximations to improve the convergence of Newton's method.
Let's say, for example, we choose the following initial approximations:
- Root 1: x = -3
- Root 2: x = 0
- Root 3: x = 1.5
4. Apply Newton's method:
Newton's method is an iterative numerical method to refine an initial approximation to find a root of an equation. Here's the general algorithm:
- Start with an initial approximation x_0.
- Calculate the next approximation x_1 using the formula:
x_1 = x_0 - f(x_0) / f'(x_0), where f'(x) represents the derivative of f(x).
- Repeat the process by substituting x_1 back into the formula to find x_2, and continue until convergence is achieved.
Specifically, for our equation f(x) = x^6 - x^5 - 6x^4 - x^2 + x + 8, we need to find the derivative f'(x).
5. Calculate the derivative:
f'(x) = 6x^5 - 5x^4 - 24x^3 - 2x + 1
6. Apply Newton's method:
- For each root, start with the chosen initial approximation and use the formula to iterate until convergence is achieved.
For Root 1 (x = -3):
- Set x_0 = -3.
- Plug into the formula: x_1 = x_0 - f(x_0) / f'(x_0)
- Repeat until convergence is achieved.
For Root 2 (x = 0):
- Set x_0 = 0.
- Plug into the formula: x_1 = x_0 - f(x_0) / f'(x_0)
- Repeat until convergence is achieved.
For Root 3 (x = 1.5):
- Set x_0 = 1.5.
- Plug into the formula: x_1 = x_0 - f(x_0) / f'(x_0)
- Repeat until convergence is achieved.
Keep iterating until the desired accuracy is reached, usually when the difference between consecutive approximations is sufficiently small (e.g., less than 0.00000001).
7. Repeat step 6 for each initial approximation to find all the roots of the equation correct to eight decimal places.
Remember to keep track of the approximations obtained at each iteration and verify their convergence to ensure the accuracy of the results.