**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