Simplify, multiply, divide, add, and subtract rational expressions Simplify complex fractions

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)