{Use of Tech} Finding all roots Use Newton’s method to find all the roots of the following functions. Use preliminary analysis and graphing to determine good initial approximations.


f(x) = x²(x - 100) + 1

Verified step by step guidance

1. Step 1: Begin by performing a preliminary analysis of the function f(x) = x^2(x - 100) + 1. Notice that this is a cubic polynomial, which means it can have up to three real roots. Consider the behavior of the function as x approaches positive and negative infinity to understand its end behavior.
2. Step 2: Graph the function f(x) to visually identify approximate locations of the roots. Look for points where the graph crosses the x-axis, as these indicate potential roots. Use these points as initial guesses for Newton's method.
3. Step 3: Recall Newton's method formula: x_{n+1} = x_n - f(x_n) / f'(x_n). Calculate the derivative of the function, f'(x) = 3x^2 - 200x, which will be used in the iterative process.
4. Step 4: Choose an initial approximation (x_0) based on the graph for each root. Substitute x_0 into the Newton's method formula to find the next approximation, x_1. Continue this iterative process until the difference between successive approximations is sufficiently small, indicating convergence to a root.
5. Step 5: Repeat the Newton's method process for each initial approximation identified from the graph to find all roots of the function. Verify each root by substituting back into the original function to ensure f(x) is close to zero.