Remainder Theorem and Synthetic Substitution

Date: 06/17/2004 at 19:58:44
From: Marc
Subject: algebra and functions

Let P(x) be the polynomial
 
   P(x) = x^15 - 2004x^14 + 2004x^13 - ... - 2004x^2 + 2004x
 
Calculate P(2003). 

I really do not know where to start.

Date: 06/18/2004 at 00:22:38
From: Doctor Greenie
Subject: Re: algebra and functions

Hi, Marc --

The Remainder Theorem tells us that the value of P(2003) is the 
remainder you get when you divide the given polynomial by (x - 2003).

A quick exercise beginning that polynomial division process using 
synthetic division shows a pattern which makes it easy to find the 
answer to the problem without completing the synthetic division:

  2003 | 1 -2004 +2004 -2004 +2004 -2004 +2004 ...
       |    2003 -2003  2003 -2003  2003 -2003 ...
       ---------------------------------------------
         1    -1     1    -1     1    -1     1 ...

The "remainder" is +1 after division of each term with an odd power 
and -1 after division of each term with an even power.  Since the 
last term of the polynomial is an odd power, the value of P(2003) is 
+1.

I hope this helps.  Please write back if you have any further 
questions about any of this.

- Doctor Greenie, The Math Forum
  http://mathforum.org/dr.math/