Beginning Algebra
Tutorial 30: Division of Polynomials
Learning Objectives
After completing this tutorial, you should be able to:
 Divide a polynomial by a monomial.
 Divide a polynomial by a polynomial using long division.

Introduction
In this tutorial we revisit something that you may not have seen since
grade school: long division. In this tutorial we are dividing polynomials,
but it follows the same steps and thought process as when you apply it
numbers. Let's forge ahead. 
Tutorial
Divide
Polynomial Monomial

Example
1: Divide . 

*Divide EVERY term by 2x
*Simplify each term

Divide
Polynomial Polynomial
Using Long Division

Step 1: Set up the long division.
The divisor (what you are dividing by) goes on the outside of the box.
The dividend (what you are dividing into) goes on the inside of the box.
When you write out the dividend, make sure that you insert 0's for any
missing terms. For example, if you had the polynomial ,
the first term has degree 4, then the next highest degree is 1. It is missing degrees 3 and 2.
So if we were to put it inside a division box, we would write it like this:
This will allow you to line up like terms when you go through the problem. 

Step 2: Divide 1st term of dividend by first
term of divisor to get first term of the quotient. 
The quotient (answer) is written above the division box.
Make sure that you line up the first term of the quotient with the term
of the dividend that has the same degree. 
Step 3: Take the term found in step 1 and multiply
it times the divisor. 
Make sure that you line up all terms of this step with the term of
the dividend that has the same degree. 
Step 4: Subtract this from the line above. 
Make sure that you subtract EVERY term found in step 3, not just the
first one. 
Step 5: Repeat until done. 
Step 6: Write out the answer. 
Your answer is the quotient that you ended up with on the top of the
division box.
If you have a remainder, write it over the divisor in your final answer. 
Example
2: Divide . 
Note that the "scratch work" that you see
at the right of the long division shows you how that step is filled in.
It shows you the "behind the scenes" of how each part comes about. 
We keep going until we can not divide anymore. It looks
like we can go one more time on this problem. We just follow the the same steps 2  4 as shown above. Our "new
divisor" is the last line 8x + 1. 
Example
3: Divide . 
We keep going until we can not divide anymore.
We just follow the the same steps 2  4 as shown above. Our "new
divisor" is always going to be the last line that was found in step 4. 
The following is the scratch work (or behind
the scenes if you will) for the rest of the problem. You
can see everything put together following the scratch work under "putting
it all together". This is just to show you how the different pieces
came about in the final answer. When you work a problem like this,
you don't necessarily have to write it out like this. You can have
it look like the final product shown after this scratch work. 
Scratch work for steps 2, 3, and 4
for the last three terms of the quotient
2nd term:
3rd term:
4th term:


Putting it all together:

Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1c: Divide.
Need Extra Help on these Topics?
Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001  2011, WTAMU and Kim Seward. All rights reserved.

