This document discusses rational expressions and operations involving them. It defines rational expressions as the quotient of two polynomials where the denominator is not equal to 0. It explains how to find the domain of a rational expression by setting the denominator equal to 0 and solving. The document provides examples of multiplying, dividing, adding and subtracting rational expressions by treating them as fractions and using properties of fractions. It emphasizes writing rational expressions in lowest terms and finding the lowest common denominator when adding or subtracting.

1. 0.5 Rational Expressions
Chapter 0 Review of Basic Concepts

2. Concepts & Objectives
⚫ Rational expressions
⚫ Finding the domain of a rational expression
⚫ Writing rational expressions in lowest terms
⚫ Multiplying and dividing rational expressions
⚫ Adding and subtracting rational expressions
⚫ Simplifying complex fractions

3. Rational Expressions
⚫ The quotient of two polynomials P and Q, with Q ≠ 0, is
called a rational expression.
⚫ The domain of a rational expression is the set of real
numbers for which the expression is defined.
⚫ Because the denominator of a rational expression
cannot be 0, the domain consists of all real numbers,
except those which make the denominator 0.
⚫ We can find these numbers by setting the
denominator equal to 0 and solving the resulting
equation.

4. Rational Expressions (cont.)
⚫ Example: What is the domain of the rational expression?
6
2
x
x
+
+

5. Rational Expressions (cont.)
⚫ Example: What is the domain of the rational expression?
Set the denominator equal to 0 and solve:
6
2
x
x
+
+
2 0x + =
2x = −

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
−
−

7. 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
−
=
+ −

8. 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
2k
=
+

9. 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.
To determine the domain, we use the original
denominator:
2
3 6
4
k
k
−
−
( )
( )( )
3 2
2 2
k
k k
−
=
+ −
3
2k
=
+
( )( )2 2 0
2
k k
k
+ − =
= 

10. 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.
To determine the domain, we use the original
denominator:
2
3 6
4
k
k
−
−
( )
( )( )
3 2
2 2
k
k k
−
=
+ −
3
2k
=
+
( )( )2 2 0
2
k k
k
+ − =
=   domain: | 2k k  

11. Multiplying and Dividing
⚫ We multiply and divide rational expressions using
definitions from ealier 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
 = = 

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
+ − +

− −

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
= =

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
= =

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
− + −
=
− + +

16. 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
−
=
+

17. 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
−
=
+

18. 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
− + −
=
− +

19. 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 3
6
p p−
=

20. 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
−
− =

21. 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
−
− =

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

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

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

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

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

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

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

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

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

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

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

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

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

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

36. 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
−
+

37. 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.

38. 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
−
+
55 66
5 5
1 1
k
k kk
k
k
k k
   −−      = =  
   + +    
Multiply both numerator and
denominator by the overall LCD, k.

39. 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
−
+
55 66
5 5
1 1
k
k kk
k
k
k k
   −−      = =  
   + +    
6 5
5
k
k
−
=
+
Multiply both numerator and
denominator by the overall LCD, k.

40. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+
( )( )LCD 1 1x x x= + −

41. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+
( )
( )
( )
( )
( )( )
( )( )
( )
( )
1 11 1
1 1 1 1
1 1 11 1
1 1 1 1
x x x x
x x x x x x
x x x x
x x x x x x
   − +
−   
+ − − +   =
   − + −
+   
− + + −   

42. Complex Fractions (cont.)
⚫ Examples (cont.)
b)
1 1
1 1
1 1
1
x x
x x
−
+ −
+
+
( )
( )
( )
( )
( )( )
( )( )
( )
( )
1 11 1
1 1 1 1
1 1 11 1
1 1 1 1
x x x x
x x x x x x
x x x x
x x x x x x
   − +
−   
+ − − +   =
   − + −
+   
− + + −   
( ) ( )
( )( )
( )( ) ( )
( )( )
1 1
1 1
1 1 1
1 1
x x x x
x x x
x x x x
x x x
− − +
− +
=
− + + −
− +

43. Complex Fractions (cont.)
⚫ Examples (cont.)
b) ( )
( )
2 2
2
2 2
2
1
1
1
x x x x
x x
x x x
x x
− − −
−
=
− + −
−

44. Complex Fractions (cont.)
⚫ Examples (cont.)
b) ( )
( )
2 2
2
2 2
2
1
1
1
x x x x
x x
x x x
x x
− − −
−
=
− + −
−
2
2
2 1
x
x x
−
=
− −

45. Classwork
⚫ 0.5 Assignment – Pg. 50: 12-36 (x4); pg. 41: 28-68 (x4);
pg. 31: 78-92 (even)
⚫ 0.5 Classwork Check (due 9/8)
⚫ Quiz 0.4 (due 9/8)