Polynomial Division Compared with Long Division

Date: 02/01/2009 at 18:48:15
From: Rashad
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

Hi Rashad,

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 + 2*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 + 2x + 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/