Example 4.  Use the Horner's method with vector coefficients to calculate and for the polynomial
        .  
Use these values to perform one step in Newton's method for approximating a root using the initial approximation  .  

Solution 4.

Use the recursive formulas to compute the sequence  .  

Use the recursive formulas to compute the sequence  .  

The coefficients and can be used with to perform one step of Newton iteration for finding a root starting with the initial guess   and using the Newton-Raphson formula:

    .

We are done.

Aside.  We can check out the formulas

    
and
    

(c) John H. Mathews 2004