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### Synthetic Division

```
Date: 11/13/97 at 14:46:29
From: Katherine Stuckey
Subject: Why does Synthetic Division Work?

Dear Dr. Math,

Could you prove to me why synthetic division works?

Katie and Jason
```

```
Date: 11/13/97 at 15:32:24
From: Doctor Jerry
Subject: Re: Why does Synthetic Division Work?

Hi Katie and Jason,

Synthetic division is an efficient arrangement of the arithmetic
required to divide a polynomial by the monomial x-a. One can do this
division by the standard procedure for dividing one polynomial by
another, but since one of the polynomials is simple, that is, is 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 mult to get 4(4-5)
1 mult to get 4(4(4-5)+1)
1 final add to finish.  The result, -18.

Okay, 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 on
computer or human time.

The better way is synthetic division, although for hand use it is
arranged a little differently. 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 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/
```
Associated Topics:
High School Number Theory
Middle School Division

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