Question

Use newton's method to approximate all real roots of x^4 - x - 2 = 0 to three consistent decimal places. (enter your answers as a comma-separated list.)

a) -1.769
b) -0.879
c) 1.246
d) 1.402

Answer 1

The approximate real roots of the equation x⁴ - x - 2 = 0 obtained using Newton's method is -1.516.

None of the given options is correct

Step-by-step explanation:

To approximate the real roots of the equation x⁴ - x - 2 = 0 using Newton's method, we can follow these steps:

1. Start by choosing an initial guess for the root. Let's say we start with x = -2.

2. Calculate the derivative of the equation, which is 4x³ - 1.

3. Use the formula for Newton's method: x1 = x₀ - f(x₀)/f'(x₀), where x1 is the updated guess, x₀ is the previous guess, f(x₀) is the value of the equation at x₀, and f'(x₀) is the value of the derivative at x₀.

4. Substitute the initial guess into the equation to find f(x₀). In this case,
`f(-2) = (-2)^4 - (-2) - 2 = 16 + 2 - 2 = 16.`

5. Substitute the initial guess into the derivative to find f'(x₀). In this case,
`f'(-2) = 4(-2)^3 - 1 = -31.`

6. Plug the values of f(x₀) and f'(x₀) into the formula: x1 = -2 - 16/(-31) = -2 + 16/31.

7. Round x₁ to three decimal places: -2 + 16/31 = -1.516.

None of the given options is correct