6.6 Divide Polynomials
Learning Objectives
By the end of this section, you will be able to:
 Divide a polynomial by a monomial
Divide a Polynomial by a Monomial
In the last chapter, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.
The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.
The sum,  , 
simplifies to  . 
Now we will do this in reverse to split a single fraction into separate fractions.
We’ll state the fraction addition property here just as you learned it and in reverse.
Fraction Addition
If , and are numbers where , then
We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.
For example,  
can be written  . 
We use this form of fraction addition to divide polynomials by monomials.
Division of a Polynomial by a Monomial
To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
EXAMPLE 1
Find the quotient: .
Divide each term of the numerator by the denominator.  
Simplify each fraction. 
TRY IT 1.1
Find the quotient: .
Show answer
TRY IT 1.2
Find the quotient: .
Show answer
Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.
EXAMPLE 2
Find the quotient: .
Rewrite as a fraction.  
Divide each term of the numerator by the denominator.  
Simplify. 
TRY IT 2.1
Find the quotient: .
Show answer
TRY IT 2.2
Find the quotient: .
Show answer
When we divide by a negative, we must be extra careful with the signs.
EXAMPLE 3
Find the quotient: .
Divide each term of the numerator by the denominator.  
Simplify. Remember, subtracting a negative is like adding a positive! 
TRY IT 3.1
Find the quotient: .
Show answer
TRY IT 3.2
Find the quotient: .
Show answer
EXAMPLE 4
Find the quotient: .
Separate the terms.  
Simplify. 
TRY IT 4.1
Find the quotient: .
Show answer
TRY IT 4.2
Find the quotient: .
Show answer
EXAMPLE 5
Find the quotient: .
Rewrite as a fraction.  
Separate the terms.  
Simplify. 
TRY IT 5.1
Find the quotient: .
Show answer
TRY IT 5.2
Find the quotient: .
Show answer
EXAMPLE 6
Find the quotient: .
Separate the terms.  
Simplify. 
TRY IT 6.1
Find the quotient: .
Show answer
TRY IT 6.2
Find the quotient: .
Show answer
EXAMPLE 7
Find the quotient: .
Separate the terms.  
Simplify. 
TRY IT 7.1
Find the quotient: .
Show answer
TRY IT 7.2
Find the quotient: .
Show answer
Access these online resources for additional instruction and practice with dividing polynomials:
Key Concepts
 Fraction Addition
 If , and are numbers where , then
and
 If , and are numbers where , then
 Division of a Polynomial by a Monomial
 To divide a polynomial by a monomial, divide each term of the polynomial by the monomial.
Practice Makes Perfect
Dividing Polynomial by Monomial
In the following exercises, divide each polynomial by the monomial.
1.  2. 
3.  4. 
5.  6. 
7.  8. 
9.  10. 
11.  12. 
13.  14. 
15.  16. 
17.  18. 
19.  20. 
21.  22. 
23.  24. 
25.  26. 
27.  28. 
29.  30. 
31.  32. 
Everyday Math
33. Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression , where represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting? 
34. Average cost Pictures Plus produces digital albums. The company’s average cost (in dollars) to make albums is given by the expression .

Writing Exercises
35. Divide and explain with words how you get each term of the quotient.  36. James divides by 6 this way: . What is wrong with his reasoning? 
Answers
1.  3.  5. 
7.  9.  11. 
13.  15.  17. 
19.  21.  23. 
25.  27.  29. 
31.  33. 45  35. Answers will vary. 
Attributions
This chapter has been adapted from “Divide Polynomials” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne AnthonySmith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.