Example 1.  Use Newton's method to find the three roots of the cubic polynomial  .  
Determine the Newton-Raphson iteration formula    that is used.  Show details of the computations for the starting value  .

Solution 1.

Graph the function.

How many real roots are there ?  Really !

The Newton-Raphson iteration formula  g[x]  is

Starting with  , Use the Newton-Raphson method to find a numerical approximation to the root.  First, do the iteration one step at a time.  Type each of the following commands in a separate cell and execute them one at a time.

From the second graph we see that there are two other real roots, use the starting values  0.0  and  1.4  to find them.
First, use the starting value .  

Then use the starting value .  

Compare our result with Mathematica's built in numerical root finder.

This can also be done with Mathematica's built in symbolic solve procedure.

(c) John H. Mathews 2004