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### 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.

```

```
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.

----------------------------------
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).

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

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