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Beginning Algebra
Tutorial 30: Division of Polynomials


 

deskLearning Objectives


After completing this tutorial, you should be able to:
  1. Divide a polynomial by a monomial.
  2. Divide a polynomial by a polynomial using long division.




desk 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.

 

 

desk Tutorial



 

 Divide

Polynomial divisionMonomial


 
Step 1: Use distributive property to write every term of the numerator over the monomial in the denominator.
 
If you need a review on the distributive property, go to Tutorial 8: Properties of Real Numbers.

 

Step 2: Simplify the fractions.
 

If you need a review on simplifying fractions, go to Tutorial 3: Fractions.

 
 
 
notebook Example 1: Divide example 1a.

 
Step 1: Use distributive property to write every term of the numerator over the monomial in the denominator

AND

Step 2: Simplify the fractions.


 
example 1b
*Divide EVERY term by 2x
 

*Simplify each term
 


 
 
 
 
 Divide

Polynomial divisionPolynomial

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 long division, 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:

long division

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.


 
 
 
notebook Example 2: Divide example 2a.

 
 

 
example 2b

 
 

 
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. 

 
example 2d
Scratch work:
example 2c

 

 
example 2f
Scratch work:
example 2e

 
 

 
example 2h
Scratch work:
example 2g

 
 

 
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 2j
Scratch work:
example 2i

 

 
example 2l
Scratch work:
example 2k

 

 
example 2n
Scratch work:
example 2m

 

 
example 2o

 
 
 
 
notebook Example 3: Divide example 3a.

 
 

 
example 3b

 
 

 
example 3d
Scratch work:
example 3c

 

 
example 3f
Scratch work:
example 3e

 
 

 
example 3g
Scratch work:
example 3f

 
 

 
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.


 

AND

Step 3 (repeated):  Take the term found in step 1 and multiply it times the divisor.

AND

Step 4 (repeated):  Subtract this from the line above.

 
  
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:
example 3h

example 3k

example 3l
 

3rd term:
example 3i

example 3m

example 3n

4th term:
example 3j

example 3o

example 3p


 
Putting it all together:
 

example 3q


 

 
example 3r

 

 

desk 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.

 

pencil Practice Problems 1a - 1c: Divide.

 


 


 

desk Need Extra Help on these Topics?


 

The following are webpages that can assist you in the topics that were covered on this page: 
 

http://www.purplemath.com/modules/polydiv.htm
This webpage helps you with dividing polynomials.

http://www.sosmath.com/algebra/factor/fac01/fac01.html 
This webpage will help you with long division. 


 

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.


 

 

 

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Last revised on August 2, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.