# Ken Ward's Mathematics Pages

### Horner's Method or Scheme

Horner's method has a variety of uses, and saves work when evaluating polynomials. It is sometimes called synthetic division.
We proceed by example:+Suppose we have the following equation:f(x)=x3+4x2+x-6
We wish to check whether -3 is a root of that equation, that is, to find f(-3). Horner's method has the advantage that fewer calculations are required. It also has the advantage of finding the reduced equation (that is by dividing f(x) by x+3).
First werite down the coefficients of the terms:
 1 4 1 -6

Next write the number to be evaluated, -3, as shown, and write 0 below the first coefficient. Add the first coefficient, 1, to 0, and write the result, 1 below:
 1 4 1 -6 -3 0 1
Multiply the 1 by -3 and write the result below 4, and add (-3+4=1):
 1 4 1 -6 -3 0 -3 1 1

Continue in this fashion until the cells are filled. The last cell below is 0, so -3 is a root of the equation.
 1 4 1 -6 -3 0 -3 -3 6 1 1 -2 0
Furthermore, the last line shows the coefficients of the equation obtained by dividing f(x) by x+3 (1,1,-2), that is 1x2+1x-2.
It is possible to continue to check whether the claim that -2 is a root or not:
 1 4 1 -6 -3 0 -3 -3 6 1 1 -2 0 -2 0 -2 2 1 -1 0
We fill in the table as before, and because we end with 0, we are assured that -2 is a root of the equation. As before, the result of dividing by x+2 is the last line (1,-1), or 1x-1, which tells us that the final root is x=1.
If the last cell isn't 0, then there is a remainder on dividing by the number.

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: