Graphical Application of Horner's Method

This Demonstration shows Horner's method for calculating the value of a polynomial at a given value of .
Horner's method reduces the number of multiplications and results in greater numerical stability by potentially avoiding the subtraction of large numbers. It is based on successive factorization to eliminate powers of greater than 1.
Suppose ; then the method rewrites .
To compute , you calculate the numbers , , …, , , :
,
,
…,
,
.
Then .
Factor out from to get .
You can select the degree of the polynomial ; the value ; and the number of steps in which is replaced by . The table shows the new coefficients after each step.
To carry out Horner's method graphically, put the points , , … , on the axis, where .
The ordinates of the blue points are calculated by a formula given for the second point from the top:
.
Suppose that the ordinates of the blue points are , ..., ; then the green points have ordinates ,..., , .
The following equalities hold:
,
,
…,
,
…,
,
.
Finally, using induction, .

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Reference
[1] D. Kurepa, Higher Algebra, Book 1 (in Croatian), Zagreb: Skolska knjiga, 1965 pp. 1070–1071.
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