Polynomial Long Division

Date: 12/03/2001 at 11:54:18
From: Rachel
Subject: Dividing a Polynomial by a Polynomial using long division

 b^9+6b^6+b^4+9b^3+4b+8 by b^3+4

Why in some questions do you need to add place holders? It has 
something to do with ascending and descending powers.

I am homeschooled and someone said that this was a great site.

Please help!

Date: 12/03/2001 at 20:14:23
From: Doctor Rick
Subject: Re: Dividing a Polynomial by a Polynomial using long division

Hi, Rachel.

In ordinary long division, you want to be sure to keep the columns 
lined up correctly - thousands, hundreds, tens, ones - and place 
holders (0 digits) help you do this. With numbers, such as 203, you 
need the zero to make it clear that the 2 is in the hundreds place; if 
you didn't write zeros, "23" could mean 23 or 203 or 230.

In polynomial long division, place holders aren't quite so critical, 
because the power of the variable is written explicitly. It's like 
writing the number 203 as 2*10^2 + 3; this is perfectly clear without 
writing 2*10^2 + 0*10 + 3. What you MUST do when doing polynomial long 
division is to order the terms from highest power to lowest power, 
just as the digits in a number are ordered from highest power of 10 to 
lowest. Place holders, while not absolutely necessary, are very useful 
because they save room for a power of the variable that might appear 
as you work, where it was previously missing. 

Let's look at your example.

          ----------------------------------
  b^3 + 4 ) b^9 + 6b^6 + b^4 + 9b^3 + 4b + 8

If we include all the place holders, we get:

                                 1b^6+0b^5+0b^4+2b^3+0b^2+1b+1
                ----------------------------------------------
 1b^3+0b^2+0b+4 ) 1b^9+0b^8+0b^7+6b^6+0b^5+1b^4+9b^3+0b^2+4b+8
                  1b^9+0b^8+0b^7+4b^6
                  -------------------
                                 2b^6+0b^5
                                 0b^6+0b^5
                                 ---------
                                 2b^6+0b^5+1b^4
                                 0b^6+0b^5+1b^4
                                 --------------
                                 2b^6+0b^5+1b^4+9b^3
                                 2b^6+0b^5+0b^4+8b^3
                                 -------------------
                                           1b^4+1b^3+0b^2
                                           0b^4+0b^3+0b^2
                                           --------------
                                           1b^4+1b^3+0b^2+4b
                                           1b^4+0b^3+0b^2+4b
                                            ----------------
                                                1b^3+0b^2+0b+8
                                                1b^3+0b^2+0b+4
                                                 -------------
                                                             4

The only place holders that you really need are those in the dividend, 
because you need to reserve space for all the columns. You don't 
really need place holders in the divisor. In the partial products, you 
don't need to write the place holders, just remember to put each term 
in the correct column for its power. 

You can also learn to skip terms in the quotient. For instance, look 
at the partial remainder of 2b^6. The next term in the quotient must 
be a b^3 term in order for the highest term of the partial product to 
be b^6. Therefore you can bring down the three terms 0b^5+1b^4+9b^3 
all at once, and skip the b^5 and b^4 terms in the quotient. You 
probably do something similar in regular long division.

Once you understand how polynomial division works, you can write it 
like this:

                         b^6         +2b^3     + b+1
        --------------------------------------------
  b^3+4 ) b^9+0b^8+0b^7+6b^6+0b^5+b^4+9b^3+0b^2+4b+8
          b^9          +4b^6
          ------------------
                        2b^6     +b^4+9b^3
                        2b^6         +8b^3
                        ------------------
                                  b^4+ b^3     +4b
                                  b^4          +4b
                                  ----------------
                                       b^3        +8
                                       b^3        +4
                                       -------------
                                                   4

Do you see how it works now, and why place holders are helpful? Use 
the place holders at least until you really understand what you're 
doing. If you aren't sure you do it correctly, please send me a 
worked-out example like mine (preferably one that you know is wrong). 
Seeing your work can help me figure out how to help you.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/