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### Polynomial Division Compared with Long Division

```Date: 02/01/2009 at 18:48:15
Subject: Dividing polynomials

I can't seem to grasp the concept!

```

```
Date: 02/02/2009 at 08:04:11
From: Doctor Ian
Subject: Re: Dividing polynomials

Suppose we have a problem like

_____
23 ) 4899

We can use "long division" like so:

213
_____
23 ) 4899
46
---
29
23
---
69
69
--
0

Does that look familiar?  I think it's a little easier to see what's
going on if we don't hide all the zeros:

3
10
200
_____
23 ) 4899
4600      <-  23 * 200
---
299
230      <-  23 *  10
---
69
69      <-  23 *   3
--
0

Now, what if we write it like this?

2*100 + 1*10 + 3*1
____________________________
2*10 + 3 ) 4*1000 + 8*100 + 9*10 + 9*1
4*1000 + 6*100
--------------
2*100 + 9*10
2*100 + 3*10
------------
6*10 + 9*1
6*10 + 9*1
----------
0

Same thing as before, but now it's more explicit about showing what's
going on.  It's even more explicit if we write it this way:

2*10^2 + 1*10^1 + 3*10^0
__________________________________
2*10^1 + 3 ) 4*10^3 + 8*10^2 + 9*10^1 + 9*10^0
4*10^3 + 6*10^2
--------------
2*10^2 + 9*10^1
2*10^2 + 3*10^1
--- ------------
6*10^1 + 9*10^0
6*10^1 + 9*10^0
---------------
0

So far, so good?  Now, suppose that, instead of 10 as the base of our
exponents, we just have some unknown number x:

2*x^2 + 1*x^1 + 3*x^0
_____________________________
2*x^1 + 3 ) 4*x^3 + 8*x^2 + 9*x^1 + 9*x^0
4*x^3 + 6*x^2
--------------
2*x^2 + 9*x^1
2*x^2 + 3*x^1
---------------
6*x^1 + 9*x^0
6*x^1 + 9*x^0
-------------
0

But that's just a polynomial division, isn't it?  If we leave out the
explicit * symbols, and ^1's, and x^0's, we have something that
should look like the kind of problem you're trying to solve:

2x^2 + 1x + 3
_____________________
2x + 3 ) 4x^3 + 8x^2 + 9x + 9
4x^3 + 6x^2
-----------
2x^2 + 9x
2x^2 + 3x
-------------
6x + 9
6x + 9
------
0

But as you can see, it's really just long division, using x instead of
10 as the base.

There are some subtleties that occur with polynomials (e.g., you can
have negative coefficients), but can you at least "grasp the concept"
now?

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Polynomials
Middle School Division

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