Let the polynomial [Graphics:Images/HornerMod_gr_1.gif] of degree n have coefficients [Graphics:Images/HornerMod_gr_2.gif].  Then [Graphics:Images/HornerMod_gr_3.gif] has the familiar form  

        [Graphics:Images/HornerMod_gr_4.gif]

Horner's method (or synthetic division) is a technique for evaluating polynomials.  It can be thought of as nested multiplication.  For example, the fifth-degree polynomial

        [Graphics:Images/HornerMod_gr_5.gif]

can be written in the "nested multiplication" form

        [Graphics:Images/HornerMod_gr_6.gif].  

Exploration

[Graphics:../Images/HornerMod_gr_7.gif]



[Graphics:../Images/HornerMod_gr_8.gif]



[Graphics:../Images/HornerMod_gr_9.gif]



[Graphics:../Images/HornerMod_gr_10.gif]



[Graphics:../Images/HornerMod_gr_11.gif]



[Graphics:../Images/HornerMod_gr_12.gif]



[Graphics:../Images/HornerMod_gr_13.gif]



[Graphics:../Images/HornerMod_gr_14.gif]



[Graphics:../Images/HornerMod_gr_15.gif]



[Graphics:../Images/HornerMod_gr_16.gif]



[Graphics:../Images/HornerMod_gr_17.gif]



[Graphics:../Images/HornerMod_gr_18.gif]



[Graphics:../Images/HornerMod_gr_19.gif]



[Graphics:../Images/HornerMod_gr_20.gif]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) John H. Mathews 2004