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.

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Graph the function.

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How many real roots are there ?  Really !

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The Newton-Raphson iteration formula  g[x]  is

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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.

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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 .

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Then use the starting value .

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

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

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(c) John H. Mathews 2004