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### Division of Unknown Polynomials

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Date: 3/18/96 at 15:6:36
From: Anonymous
Subject: Algebra problem

I am an eighth grade student at the Willmar Junior High School in
Willmar, Minnesota.  I have a math question that I need help on.

When a polynomial P(x) is divided by x-1, the remainder is 3.
When P(x)is divided by x-2, the remainder is 5.  Find the
remainder when P(x) is divided by x^2-3x+2.

Thank you,
Jacob
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Date: 3/22/96 at 19:28:51
From: Doctor Steven
Subject: Re: Algebra problem

This is a tough problem for an eighth grade class!

First let's look at what we know:

1. We know P(x) when divided by (x-1) gives a remainder of 3.
So P(x) = f(x)*(x-1) + 3.  (We don't care what f(x) is)

2. We know P(x) when divided by (x-2) gives a remainder of 5.
So P(x) = g(x)*(x-2) + 5, also.(We don't care what g(x) is
either!)

So P(x) = f(x)*(x-1) + 3 = g(x)*(x-2) + 5.

Subtract 5 from every part of this three part equation to get:
P(x) - 5 = f(x)*(x-1) - 2 = g(x)*(x-2) + 0.

Well, we also know that (x-1) = 1*(x-2) + 1. So the remainder of
x-1 when divided by x-2 is +1.

Now check this out for some higher mathematics (really though it's
pretty easy):
____                                    ___
Look at 4| 5   its remainder is 1.  Now look at 4| 3  its
remainder is 3.
____
Now look at 4| 15  its remainder is 3, or 3*1 the product
of the remainders of 5/4 and 3/4.  This works for anything
that is multiplied together.

So f(x)*(x-1) - 2 better not have a remainder when divided by
(x-2) (since it equals the far righthand side, which definitely
doesn't have a remainder when divided by (x-2) ).

This tells us that f(x) better have a remainder of 2 when
divided by (x-2) (since rem(2)*rem(1) - rem(2) = rem(0) ).

So f(x) = m(x)*(x-2) + 2. (We don't care what m(x) is)

Now we have:

P(x) - 5 = [m(x)*(x-2) + 2](x-1) - 2 = g(x)(x-2) + 0.

But all we need to look at now is:

P(x) - 5 = [m(x)*(x-2) + 2](x-1) - 2.

Add 5 to both sides to get:

P(x) = [m(x)*(x-2) + 2](x-1) + 3.

Then simplify on the right to get:

P(x) = m(x)*(x-2)*(x-1) + (2x - 2) + 3.

Simplify some more to get:

P(x) = m(x)*(x^2 - 3x + 2) + 2x + 1.

So the remainder of P(x) when divided by x^2 - 3x + 2 is
(2x + 1)

Whew! That was a toughy. ;)

-Doctor Steven,  The Math Forum

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Associated Topics:
High School Polynomials

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