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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 

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)+1
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


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

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
Associated Topics:
High School Number Theory
Middle School Division

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