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Synthetic Division

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Date: 08/05/97 at 00:49:36
From: Barry Slator
Subject: Synthetic division (algebra)

Dear Dr. Math,

I need you to explain to me synthetic division.  I am currently
preparing for math during the summer. I ran into synthetic
division, and I do not quite understand.  If you could explain to me
how it works and give an example it would be greatly appreciated!
Thank you !

Sincerely,

Barry Slator
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```
Date: 08/05/97 at 08:02:01
From: Doctor Jerry
Subject: Re: Synthetic division (algebra)

Hi Barry,

Synthetic division is an efficient arrangement of the arithmetic
required to divide a polynomial by the monomial x-a. Although one
can do this by long division, because the divisor is the simple
polynomial x-a the work can be shortened.

If f(x) = A*x^3+B*x^2+C*x+D, for example, synthetic division also
provides an efficient way of calculating f(a). This is probably the
more useful way of looking at synthetic division.

I'll give an illustration.

Suppose f(x) = x^3-5x^2+2x-10.  If we want to calculate f(4), we may
do this:

f(4) = 4^3-5*4^2+2*4-10.

Let's count the number of multiplications and additions required.

2 mults to get 4^2
1 more mult to get  4^3
1 mult to get 5*4^2
1 mult to get 2*4
3 adds to get 4^3-5*4^2+2*4-10 = -18

There's a better way. We write f(x) = x(x(x-5)+2)-10, which is called
nested multiplication. Now count again.

1 add to get 4-5
1 mult to get 4(4-5)
1 add to get 4(4-5)+2
1 mult to get 4(4(4-5)+2)
1 final add to finish.  The result, -18.

That's 2 mults and 3 adds, compared to 5 mults and 3 adds above.
For higher-degree polynomials, this can add up to big savings in
computer or human time.

So, the better way for evaluation is synthetic division. For doing
it by hand the nested arithmetic can be done in a different format.
You write

1   -5   2    -10 |  4

and then draw a line and bring down the first coefficient.

1   -5    2   -10 |  4

____________________
1

After that, you repeatedly multiply by 4 and add to the top line.
I'll give the result.

1   -5    2    -10 |  4
4   -4    - 8
____________________
1   -1   -2    -18

If you look at x(x(x-5)+2)-10, with x = 4, you'll see that the
arithmetic matches synthetic division.

Finally, one frequent use of synthetic division is to test numbers to
see if they are roots of a polynomial. The number 4 is not a root
since the last number generated (-18 in this case) is not zero. If you
get 0, then the number  tried is a root, assuming you didn't make any
mistakes in arithmetic.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Basic Algebra
High School Polynomials

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