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This message was updated on 8/28/2002 11:38:09 AM by Soroban

Math Trick: Evaluating a Polynomial
posted on: 8/28/2002 9:40:39 AM

Given: f(x) = 2x^4 - 4x^3 + 3x^2 - 2x + 6.
Find f(5).

It is a typical algebra problem,
demanding that we find 5^2, 5^3, and 5^4
and that we perform the indicated arithmetic:

f(5) = 2(625) - 4(125) + 3(25) - 2(5) + 6
= 1250 - 500 + 75 - 10 + 6 = 821

There is, however, a lesser-known and highly efficient shortcut,
requiring no exponents and perhaps a fifth of the time.

First, consider only the coefficients of the polynomial:
(2,-4, 3,-2, 6), which I will call "terms".
We begin with the first term, 2.

1. Multiply by 5: 2 x 5 = 10
2. Add the next term: 10 + (-4) = 6

Repeat steps 1 and 2.

Multiply by 5: 6 x 5 = 30
Add the next term: 30 + 3 = 33

Multiply by 5: 33 x 5 = 165
Add the next term: 165 - 2 = 163

Multiply by 5: 163 x 5 = 815
Add the last term: 815 + 6 = 821

Impressive, but was that a coincidence?
No, it is a valid procedure.

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An informal proof
Begin with f(x):2x^4 - 4x^3 + 3x^2 - 2x + 6
Factor x from the first four terms:(2x^3 - 4x^2 + 3x - 2)x + 6
Factor x from the first three terms:([2x^2 - 4x + 3]x - 2)x + 6
Factor x from the first two terms:([(2x - 4)x + 3]x - 2)x + 6
Now, let x = 5.We begin "inside" with 2 times 5, subtract 4,
then multiply that by 5, add 3,
then multiply that by 5, subtract 2,
then multiply that by 5, add 6.

We have duplicated the exact steps in the shortcut method.

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Important

On a calculator, press "=" after each addition or subtraction.

We have: 2*5 - 4 and we want to multiply that (6) by 5.
The keystrokes are: [2][x][5][-][4][=], then [x][5], etc.
Without the "=", the calculator would find 2*5 - 4*5 instead.
I call this the "Equals Rule".

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To make the procedure even smoother, here's another trick:
I call it "Drop the exponents."

To find f(5), let x = 5, but disregard all exponents.
We write: 2*5 - 4*5 + 3*5 - 2*5 + 6.
This is mathematically incorrect, but it outlines our steps:
2 times 5, subtract 4, times 5, plus 3, times 5, etc.
remembering the Equals Rule, of course.

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We can convert numbers from another base to base 10.

Consider 24301(base 5). It is, after all, a "polynomial in 5."
24301(base 5) = 2(5^4) + 4(5^3) + 3(5^2) + 0(5) + 1

We can write: 2*5 + 4*5 + 3*5 + 0*5 + 1
and will have determined the answer (1826)
while others are still counting right-to-left
to learn that a 5^4 is needed.

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