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Polynomial Problem

Date: 11 Mar 1995 16:21:18 -0500
From: Anonymous
Subject: Algebra II

Anna-Jonesboro needs your help again.  The problem is: 
A remainder of 9 results when the polynomial p(x) is 
divided by x - 2.  A remaimder of -3 results when p(x) 
is divided by x + 2.  A remainder of 3 results when p(x)
is divided by x - 1.  Find the remainder when p(x) is 
divided by (x - 2)(x + 2)(x - 1). Could you just give us a 
hint first? 
                   Ruth Roy A-J.

Date: 23 Mar 1995 00:50:39 -0500
From: Dr. Ken
Subject: Re: Algebra II

Hey folks,

It's Ethan and after a little discussion with my 
friend Ken, and a little help from a professor here we 
got it.  The idea is really neat.  Here is the hint.  I will 
start with a pretty vague hint and if you need more 
then write back.

All right we have this function and we want to know 
the remainder when we divide by that product.  I think 
that it will be easier if we just think of it as trying to 
find the smallest polynomial that we can that has the 
given properties.  That will be the remainder.  Okay so 
here is the process.  It may not make sense to you 
because I don't want to give away too much.

Let us call f(x) our function.  Well then if f(x)/x-2 has 
remainder 9 then f(2) must be nine.  Similarly f(-2) = -3 
AND F(1) = 3 I will let you think about why that is true 
since you said that you just wanted a hint.  I will also let 
you think about how we can use this fact to find the answer.

Write back if this isn't enough of a hint.
Ethan Doctor On Call

-Ken "Dr." Math
Associated Topics:
High School Basic Algebra

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