Example 8.  Use the Newton-Horner method and find all the real roots of the polynomial  
  

Solution 8.

Construct a the polynomial.

Aside.  This polynomial has known roots  .  

Find the ten roots starting from largest to smallest.  Use a maximum of 40 iterations to find each root.

Construct a the polynomial.

Find the ten roots starting from smallest to largest.  Use a maximum of 40 iterations to find each root.

Now compare the two ways of computing the roots.  For comparison purposes, list the roots in ascending order.

Looking at the above table we see that the computed roots are more accurate if they are computed in ascending order.

(c) John H. Mathews 2004