Synthetic Division

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

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/