Question

use newtons method to find the two real solutions of the equation x^4-2x^3-x^2-2x+2

Answer 1

I am not fond of Newton's method, but I know Descarte's rule of signs. The number of sign changes will tell you the the maximum possible number of positive roots.

Ex. X^4-2x^3-x^2-2x+2

+---+

There can only be up to 2 positive roots.

Answer 2

To use Newton’s method we differentiate the polynomial: 4x³-6x²-2x-2. A graph shows zeroes around 0.6 and 2.6 so we can use these as starters.

X=X-(X⁴-2X³-X²-2x+2)/(4X³-6X²-2X-2) where X is an approximation (to be substituted for X on the right side) and X on the left side is a more accurate result. We then substitute the result into the right side to get the next iteration. 0.6306049822064.. is the first iteration. Next: 0.6301155198..., then 0.6301153962, which is stable.

Now we use X=2.6. Final result is 2.573272582.