Teaching Synthetic Division

Date: 10/23/96 at 14:11:26
From: Elizabeth Kenyon
Subject: Teaching Synthetic Division

I am a secondary education mathematics major and I have a question.  
In my education computer class we are putting together projects.  Mine 
is on synthetic division.  I am really confused as to how to put 
information together to be able to teach this, on the computer, to the 
rest of my class. 

I would really appreciate it if you could help me out and give me a 
few suggestions as to how I can teach this subject clearly.

Thank You Very Much,

Elizabeth Kenyon

Date: 10/23/96 at 16:5:47
From: Doctor Jerry
Subject: Re: Teaching Synthetic Division

Although many teachers think that synthetic division is an old-
fashioned topic, it can be used to teach several ideas.  It leads to a 
very efficient method for evaluating polynomials and, in this form, is 
used often.

I have several comments or suggestions.

1. The actual algorithm is nothing more than a compression of the long 
division algorithm.  If p(x) is the polynomial, then if you divide 
p(x) by the polynomial x-a, the remainder will be a number r. This 
remainder is actually equal to p(a).  If you look at the steps in the 
long division process of getting r and then compare them to the steps 
in the synthetic division process, you will note (and the students 
will note) that the synthetic division just removes all of the extra 
steps, compressing them down to the essential arithmetic.

2. I have implemented a synthetic division algorithm on a calculator.  
It works very well.  If you have an HP48, I can give you the program.  
I don't have it for a TI.

3. To evaluate p(x) for various x values can be very time consuming.  
If p(x) is factored in a certain way, great savings are possible.  
I'll give two examples.

  if p(x) = x^2+a*x+b, then write as x(x+a)+b
  if p(x) = x^3+a*x^2+b*x+c, then write as x(x(x+a)+b)+c
etc.

The evaluation of p(y) for any number y using the factored form is 
just synthetic division!  Try it and you will notice the same 
arithmetic.

I hope my remarks have been useful to you.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/