You have indicated these coefficients, although I worry there may be issues, because of the use of lower versus upper case W, etc.
But this is a CUBIC polynomial. I would use roots or solve, but really, I fear the problem is you have a mistake somewhere in the parameters. Using a Newton method to solve for the roots of a cubic polynomial may be the wrong idea (unless it is indicated as part of your homework assignment), because there is not even any assurance you will find the correct root. When a tool is a available to give you all the roots, then I would use it.
But first, let us look at the polynomial and its coefficients. I'll do this all symbolically and numerically, to learn what is happening.
A = 2.432;
E = 770000;
co = 0.0000189;
t = 10;
s = 15000;
W = 0.85;
P = x^3- x^2*((A*E)*(co*t+(W^2*s^2/24*3819^2))-3819)-(W^2*s^2*A*E/24)
x^3 - 184996171323809955840*x^2 - 6494315519999999/512
The coefficients are clearly a bit strange.
When I use roots on these coefficients, I see:
So roots tells me there is a root out at 1.85e20, but also a double root near zero. If I use solve to tell me the complete story, now I understand what happened to your Newton solve.
- 1.853138791764258719098609e-28 - 0.0002618486514794448919516663i
- 1.853138791764258719098609e-28 + 0.0002618486514794448919516663i
So there is a complex conjugate pair of roots near x==0. But the polynomial has such extreme coefficients that roots becomes confused, and the Newton method also gets lost.
There is not any root around 1000. NOT EVEN CLOSE. So Newton's method essentially found a root at/near zero. The problem is not in your Newton's method code, but in the polynomial coefficients themselves.
As such, if you think there should be a root near 1000, then you have either the wrong polynomial, or the wrong set of parameters - probably the wrong units on some parameter.
I have no idea what is the material involved here. But I will hazard a guess the problem lies in your units. Since we are not even told what this problem is intended to model, I will not try to guess which of your parameters are in error. I suppose I could do a units investigation of that formula, to decide if it makes sense. (Actually, I did try to make some sense of that you have written, but it is a bit confusing.) Even for that, I would need to know what some of the parameters are intended to mean. Is "rulling span" intended to denote a length? What is the material involved?