2
Evaluation, Division, and
Expansion
2.1 Homer’s Method
A Knight wishes to evaluate the polynomial 8t3 - 5t2 + 4t + 1 at t = 2. He
takes it to the Royal Reckoner, who charges 10 sous for each multiplication
and 5 sous for each addition. Since there are three multiplications required
for the first term, two for the second and one for the third, the multiplica-
tions will cost 60 sous. In addition, there will be a 15 sou charge for adding
the terms, for a grand total of 75 sous. The Knight wonders whether the
job could be done more cheaply.
After some thought, he makes a suggestion. Write the first two terms
in the form (8t - 5)t2, and substitute in t = 2. We then have one mul-
tiplication and one subtraction inside the bracket, followed by two other
multiplications, for a total cost of 35 sous. This compares very favorably
with the 55 sous it would have cost using the Royal Reckoner’s method.
Why not carry this regrouping further? The sum of the first three terms is
equal to
((8t - 5)t + 4)t.
We still have only three multiplications along with two additions or sub-
tractions when we substitute in t = 2 and evaluate. All we have to do
now to get the value of the polynomial we started with is to make one
more addition. The total cost is 45 sous. The nested form is going to save
money!
Exercises
- Consider the problem of evaluating the polynomial
3t3 - 4t2 + 7t + 2at t = 3.(a) Show that the polynomial can be written in the form(((3t - 4)t + 7)t + 2)and use this to effect a cheap evaluation at t = 3.