Example 1.  Use synthetic division (Horner's method) to find for the polynomial
        .  

Solution 1.

This example is for pedagogical purposes.  Our eventual goal is to use a vector for the coefficients, but for now we will enter them separately into each coefficient.  First we will check out the nested multiplication idea.  

Now we will use the recursive formulas to compute the sequence .  

Moreover, let us verify the formulas for the quotient and remainder  

          
    
        ,  

where    is the quotient polynomial of degree  n-1  and    is the remainder.  

(c) John H. Mathews 2004