Associated Topics || Dr. Math Home || Search Dr. Math

### 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

- Doctor Greenie, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search