The document outlines the essential concepts and objectives for working with rational expressions, including simplification, multiplication, division, addition, and subtraction. It explains that a rational expression is the quotient of two polynomials and provides various examples for each operation. Additionally, it discusses complex fractions and methods for simplifying them.

1. 1.6 Rational Expressions
Chapter 1 Prerequisites

2. Concepts & Objectives
⚫ Objectives for this section:
⚫ Simplify rational expressions.
⚫ Multiply and divide rational expressions.
⚫ Add and subtract rational expressions.
⚫ Simplify complex rational expressions.

3. Rational Expressions
⚫ The quotient of two polynomial expressions P and Q,
with Q ≠ 0, is called a rational expression.

4. Rational Expressions (cont.)
⚫ A rational expression is written in lowest terms when
the greatest common factor of its numerator and
denominator is 1.
⚫ Example: Write the rational expression in lowest terms.
2
3 6
4
k
k
−
−

5. Rational Expressions (cont.)
⚫ A rational expression is written in lowest terms when
the greatest common factor of its numerator and
denominator is 1.
⚫ Example: Write the rational expression in lowest terms.
2
3 6
4
k
k
−
−
( )
( )( )
3 2
2 2
k
k k
−
=
+ −

6. Rational Expressions (cont.)
⚫ A rational expression is written in lowest terms when
the greatest common factor of its numerator and
denominator is 1.
⚫ Example: Write the rational expression in lowest terms.
2
3 6
4
k
k
−
−
( )
( )( )
3 2
2 2
k
k k
−
=
+ −
3
2
k
=
+

7. Multiplying and Dividing
⚫ We multiply and divide rational expressions using
definitions from earlier work with fractions.
⚫ It is generally assumed that you will leave your answer
in lowest terms.
For fractions and , (b ≠ 0, d ≠ 0),
and
a
b
c
d
a c ac
b d bd
= ( )
0
a c a d ad
c
b d b c bc
 = = 

8. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −

9. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =

10. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =

11. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =
( )( )
( )( )
4 5 2 2 1
2 1 2 4
n n n
n n n
− + −
=
− + +

12. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =
( )( )
( )( )
4 5 2 2 1
2 1 2 4
n n n
n n n
− + −
=
− + +
4 5
4
n
n
−
=
+

13. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =
( )( )
( )( )
4 5 2 2 1
2 1 2 4
n n n
n n n
− + −
=
− + +
4 5
4
n
n
−
=
+

14. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =
( )( )
( )( )
4 5 2 2 1
2 1 2 4
n n n
n n n
− + −
=
− + +
4 5
4
n
n
−
=
+
( )( )
( )
( )
( )
3
2
3 1 4 12 2 3
8 3 1 9 4
p p p p
p p p
− + −
=
− +

15. Multiplying and Dividing (cont.)
Examples: Multiply or divide, as indicated
1. 2.
3.
2
5
2 27
9 8
y
y
2
2
4 3 10 2 1
2 3 2 4
n n n
n n n
+ − −
+ − +
2
3 2 4 3
3 11 4 9 36
24 8 24 36
p p p
p p p p
+ − +

− −
2
5 3
54 3
72 4
y
y y
= =
( )( )
( )( )
4 5 2 2 1
2 1 2 4
n n n
n n n
− + −
=
− + +
4 5
4
n
n
−
=
+
( )( )
( )
( )
( )
3
2
3 1 4 12 2 3
8 3 1 9 4
p p p p
p p p
− + −
=
− +
( ) 2
2 3 2 3
6 6
p p p
− −
= =

16. Addition and Subtraction
⚫ From earlier work with fractions:
For fractions and , (b ≠ 0, d ≠ 0),
and
a
b
c
d
+
a c ad bc
b d bd
+
=
a c ad bc
b d bd
−
− =

17. Addition and Subtraction
⚫ From earlier work with fractions:
⚫ How this applies to rational expressions is that you will
need to write each denominator as a product of prime
factors and then use that to find the lowest common
denominator (LCD) that includes all of the factors.
For fractions and , (b ≠ 0, d ≠ 0),
and
a
b
c
d
+
a c ad bc
b d bd
+
=
a c ad bc
b d bd
−
− =

18. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
1. 2.
3.
2
5 1
9 6
x x
+
8
2 2
y
y y
+
− −
( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + −

19. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
1. 2
5 1
9 6
x x
+
( )( )
2 2 1 1 1
5 1
3 3 2
x x
= +

20. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
1. 2
5 1
9 6
x x
+
( )( )
2 2 1 1 1
5 1
3 3 2
x x
= + 2 1 2 2
LCD 3 2 18
x x
= =

21. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
1. 2
5 1
9 6
x x
+
( )( )
2 2 1 1 1
5 1
3 3 2
x x
= + 2 1 2 2
LCD 3 2 18
x x
= =
( )( )
2 2
5 2 1 3
3 2 3 2 3
x
x x x
   
= +
   
   

22. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
1. 2
5 1
9 6
x x
+
( )( )
2 2 1 1 1
5 1
3 3 2
x x
= + 2 1 2 2
LCD 3 2 18
x x
= =
( )( )
2 2
5 2 1 3
3 2 3 2 3
x
x x x
   
= +
   
   
2 2 2
10 3 10 3
18 18 18
x x
x x x
+
= + =

23. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
2.
8
2 2
y
y y
+
− − ( )
8
2 2
y
y y
= +
− − −

24. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
2.
8
2 2
y
y y
+
− − ( )
8
2 2
y
y y
= +
− − − ( )
LCD 2
y
= −

25. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
2.
8
2 2
y
y y
+
− − ( )
8
2 2
y
y y
= +
− − − ( )
LCD 2
y
= −
( )
8 1
2 2 1
y
y y
−
 
= +  
− − − −
 

26. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
2.
8
2 2
y
y y
+
− − ( )
8
2 2
y
y y
= +
− − − ( )
LCD 2
y
= −
( )
8 1
2 2 1
y
y y
−
 
= +  
− − − −
 
8
2
y
y
−
=
−

27. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
3. ( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + − ( )( )( )
LCD 3 5 5
x x x
= − + −

28. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
3. ( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + − ( )( )( )
LCD 3 5 5
x x x
= − + −
( )( ) ( )( )
4 5 6 3
3 5 5 5 5 3
x x
x x x x x x
− −
   
= −
   
− + − + − −
   

29. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
3. ( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + − ( )( )( )
LCD 3 5 5
x x x
= − + −
( )( ) ( )( )
4 5 6 3
3 5 5 5 5 3
x x
x x x x x x
− −
   
= −
   
− + − + − −
   
( ) ( )
( )( )( )
4 5 6 3
3 5 5
x x
x x x
− − −
=
− + −

30. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
3. ( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + − ( )( )( )
LCD 3 5 5
x x x
= − + −
( )( ) ( )( )
4 5 6 3
3 5 5 5 5 3
x x
x x x x x x
− −
   
= −
   
− + − + − −
   
( ) ( )
( )( )( )
4 5 6 3
3 5 5
x x
x x x
− − −
=
− + −
( )( )( )
4 20 6 18
3 5 5
x x
x x x
− − +
=
− + −

31. Adding and Subtracting (cont.)
Examples: Add or subtract, as indicated.
3. ( )( ) ( )( )
4 6
3 5 5 5
x x x x
−
− + + − ( )( )( )
LCD 3 5 5
x x x
= − + −
( )( ) ( )( )
4 5 6 3
3 5 5 5 5 3
x x
x x x x x x
− −
   
= −
   
− + − + − −
   
( ) ( )
( )( )( )
4 5 6 3
3 5 5
x x
x x x
− − −
=
− + −
( )( )( ) ( )( )( )
4 20 6 18 2 2
3 5 5 3 5 5
x x x
x x x x x x
− − + − −
= =
− + − − + −

32. Complex Fractions
⚫ We call the quotient of two rational expressions a
complex fraction.
⚫ There are a couple of different ways to handle these.
⚫ Example: Simplify each complex fraction.
a)
5
6
5
1
k
k
−
+

33. Complex Fractions
⚫ We call the quotient of two rational expressions a
complex fraction.
⚫ There are a couple of different ways to handle these.
⚫ Example: Simplify each complex fraction.
a)
5
6
5
1
k
k
−
+
5
6
5
1
k k
k
k
 
−
 
 
=  
 
  
+
 
Multiply both numerator and
denominator by the overall LCD, k.

34. Complex Fractions
⚫ We call the quotient of two rational expressions a
complex fraction.
⚫ There are a couple of different ways to handle these.
⚫ Example: Simplify each complex fraction.
a)
5
6
5
1
k
k
−
+
5
5 6
6
5 5
1 1
k
k k
k
k
k
k k
 
  −
−  
 
   
= =
 
 
 
  
+ +
 
   
Multiply both numerator and
denominator by the overall LCD, k.

35. Complex Fractions
⚫ We call the quotient of two rational expressions a
complex fraction.
⚫ There are a couple of different ways to handle these.
⚫ Example: Simplify each complex fraction.
a)
5
6
5
1
k
k
−
+
5
5 6
6
5 5
1 1
k
k k
k
k
k
k k
 
  −
−  
 
   
= =
 
 
 
  
+ +
 
   
6 5
5
k
k
−
=
+
Multiply both numerator and
denominator by the overall LCD, k.

36. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+

37. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+
1 1 1 1
1 1 1 1
1 1 1
1 1
x x
x x x x
x x
x x x x
− +
     
−
     
− + + −
     
=
+
     
+
     
+ +
     
Use the LCDs of the
numeration and
denominator to
rewrite them as
single terms.

38. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+
1 1 1 1
1 1 1 1
1 1 1
1 1
x x
x x x x
x x
x x x x
− +
     
−
     
− + + −
     
=
+
     
+
     
+ +
     
( )
( )( )
( )
( )( )
( )
1 1 2
1 1 1 1
1 2 1
1 1
x x
x x x x
x x x
x x x x
− − + −
− + − +
= =
+ + +
+ +
Use the LCDs of the
numeration and
denominator to
rewrite them as
single terms.

39. Complex Fractions (cont.)
⚫ Examples (cont.)
( )( )
( )
( )( ) ( )
2
1 1 2 2 1
2 1 1 1 1
1
x x x
x x x x x
x x
−
− + − +
= 
+ − + +
+
Then, rewrite the
complex fraction as a
division expression,
then as multiplication,
and simplify.

40. Complex Fractions (cont.)
⚫ Examples (cont.)
( )( )
( )
( )( ) ( )
2
1 1 2 2 1
2 1 1 1 1
1
x x x
x x x x x
x x
−
− + − +
= 
+ − + +
+
Then, rewrite the
complex fraction as a
division expression,
then as multiplication,
and simplify.
( )( )
( )
1
2
1 1 2 1
x x
x x x
+
−
= 
− + +

41. Complex Fractions (cont.)
⚫ Examples (cont.)
( )( )
( )
( )( ) ( )
2
1 1 2 2 1
2 1 1 1 1
1
x x x
x x x x x
x x
−
− + − +
= 
+ − + +
+
( )( ) 2
2 2
1 2 1 2 1
x x
x x x x
− −
= =
− + − −
Then, rewrite the
complex fraction as a
division expression,
then as multiplication,
and simplify.
( )( )
( )
1
2
1 1 2 1
x x
x x x
+
−
= 
− + +

42. Classwork
⚫ 1.6: 4-12 (even); 1.5: 16-36 (even); 1.4: 38-52 (even)
⚫ 1.6 Classwork Check
⚫ Quiz 1.5