Long Division of Polynomials

Date: 26 Jun 1995 11:04:20 -0400
From: beth parrill
Subject:  Long division

How do you use long division to solve polynomials with remainders?  
I am trying to help my mom figure it out for a boy that she tutors.  
He is in the 10th grade.

Thanks...   Beth

Date: 26 Jun 1995 12:19:51 -0400
From: Dr. Ken
Subject: Re: Long division

Hello there!

Here's an example of when you'd use polynomial long division.  The 
steps are quite similar to the steps you do when dividing numbers, but 
that may not be helpful until you already know how to do it.

Let's say we have the polynomial x^5 - 2x^2 + 4, and we want to divide 
it by x-4.  Set it up like this:
         __________________________________
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4

Then ask yourself "how many times does x (the highest power in the divisor)
go into x^5 (the highest power in the dividend)?"  In other words, what do
you have to multiply x by to get x^5?  Why, x^4, naturally.  So do the
multiplication (of the whole divisor) and write the factor you multiplied by
on top:

         _______x^4________________________
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
         x^5 - 4x^4

Now subtract from the dividend what you've just written down.  To do that,
I'll distribute a negative sign through what I've just written and add them:

         _______x^4________________________
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
        -x^5 + 4x^4
         ----------
               4x^4

Now bring down the next term in the dividend, and repeat the process:

         _______x^4_+_4x^3_________________
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
        -x^5 + 4x^4
         ----------
               4x^4 + 0x^3
               4x^4 - 16x^3
               ------------   (--subtract)
                     16x^3

Do it again:

         _______x^4_+_4x^3_+16x^2__________
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
        -x^5 + 4x^4
         ----------
               4x^4 + 0x^3
               4x^4 - 16x^3
               ------------ 
                     16x^3 + 2x^2
                     16x^3 - 64x^2
                     -------------  (--subtract)
                            66x^2
Again:

         _______x^4_+_4x^3_+16x^2_+66x_____
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
        -x^5 + 4x^4
         ----------
               4x^4 + 0x^3
               4x^4 - 16x^3
               ------------
                     16x^3 + 2x^2
                     16x^3 - 64x^2
                     -------------
                            66x^2 + 0x
                            66x^2 -264x
                            -----------
                                  264x
One more time:

         _______x^4_+_4x^3_+16x^2_+66x_+264___
   x-4  )x^5 + 0x^4 + 0x^3 + 2x^2 + 0x + 4
        -x^5 + 4x^4
         ----------
               4x^4 + 0x^3
               4x^4 - 16x^3
               ------------
                     16x^3 + 2x^2
                     16x^3 - 64x^2
                     -------------
                            66x^2 + 0x
                            66x^2 -264x
                            -----------
                                  264x + 4
                                  264x - 1056
                                  -----------
                                        1060

Yay!  So the answer is that x^5 + 2x^2 + 4 divided by x-4 is
x^4 + 4x^3 + 16x^2 + 66x + 264, with a remainder of 1060.

Hope that helps, and if you have any questions about this, please ask us.

-K