7.1
The Fundamental Property of Rational Expressions
1 Find the numerical value of a rational expression.
2 Find the values of the variable for which a rational expression is undefined.
3 Write rational expressions in lowest terms.
4 Recognize equivalent forms of rational expressions.

The Fundamental Property of Rational Expressions
The quotient of two integers (with the denominator not 0), such as
2
3 or is called a
4 rational number.
In the same way, the quotient of two polynomials with the denominator not equal to 0 is called a rational expression.
Rational Expression
P
A rational expression is an expression of the form where P and Q
Q are polynomials, with Q ≠ 0.
x

6 x
3 
8
, y
9 x

3
,
2 m
3
8
Examples of rational expressions

EXAMPLE 1
Evaluating Rational Expressions
Find the value of the rational expression, when x = 3.
x
2 x

1
Solution:

2
 
  
1

3

3
7

Find the values of the variable for which a rational expression is undefined.
P
In the definition of a rational expression Q cannot equal 0 .
The denominator of a rational expression cannot equal 0 because division by 0 is undefined .
For instance, in the rational expression
3 x

6
2 x

4 Denominator cannot equal 0 the variable x can take on any real number value except 2. If x is 2 , then the denominator becomes 2(2) − 4 = 0 , making the expression undefined. Thus, x cannot equal 2 . We indicate this restriction by writing x ≠ 2 .
Since we are solving to find values that make the expression undefined, we write the answer as “variable ≠ value”, not “variable = value or { } .

Find the values of the variable for which a rational expression is undefined. (cont’d)
Determining When a Rational Expression is Undefined
Step 1: Set the denominator of the rational expression equal to 0.
Step 2: Solve this equation.
Step 3: The solutions of the equation are the values that make the rational expression undefined. The variable cannot equal these values.
The numerator of a rational expression may be any real number.
If the numerator equals 0 and the denominator does not equal 0, then the rational expression equals 0.
Slide 7.1-8

EXAMPLE 2
Finding Values That Make Rational Expressions Undefined
Find any values of the variable for which each rational expression is undefined.
Solution: x

2 x

5 x 5 x

5

5

0
 x

0
5
5
3 r r
2 
6 r

8

3 r
 r

2
 r

4
 r 2 0 r

2

2

0

2 r
 
2 r 4 0 r

4

4

0

4 r
 
4
5 z

1 z
2 
5 never undefined

Write rational expressions in lowest terms.
A fraction such as is said to be in lowest terms.
3
Lowest Terms
A rational expression ( Q ≠ 0) is in lowest terms if the greatest
Q common factor of its numerator and denominator is 1.
Fundamental Property of Rational Expressions
If ( Q ≠ 0) is a rational expression and if K represents any
Q polynomial, where K ≠ 0, then
PK

P
QK Q
.
This property is based on the identity property of multiplication, since
P
Q
K
K
P
Q
K
K
P
Q
1
P
Q
.

EXAMPLE 3
Writing in Lowest Terms
Write each rational expression in lowest terms.
Solution:
15
45

3
 
5

1
3
6 p
3
2 p
2

2 3 p
 p p
2


3 p

Write rational expressions in lowest terms. (cont’d)
Writing a Rational Expression in Lowest Terms
Step 1: Factor the numerator and denominator completely.
Step 2: Use the fundamental property to divide out any common factors.
Quotient of Opposites
If the numerator and the denominator of a rational expression are x
 y opposites, as in then the rational expression is equal to −1.
Rational expressions cannot be written in lowest terms until after the numerator and denominator have been factored. Only common factors can be divided out, not common terms.
6 x


9

3
4 x 6 2


2 x

3
2 x

3



3
2
6

4 x x
Numerator cannot be factored.

EXAMPLE 4
Writing in Lowest Terms
Write in lowest terms.
4 y

2
6 y

3
Solution:

2


2 y

1
3 2 y

1



2
3 a
2  b
2 a
2 
2
 2


    a
 b
  


  a
 b


EXAMPLE 5
Writing in Lowest Terms (Factors Are Opposites) z
2


5
Write in lowest terms.
2
5 z

Solution:
1

1
 z z
2
2


5
5

 
 
1

EXAMPLE 6
Writing in Lowest Terms (Factors Are Opposites)
Write each rational expression in lowest terms.
5
 y y

5
25 x
2 
16 x

Solution:

1
 y

5
 y

5


5 x

4

3

 
5 x

4
5 x

4


 
1


5 x

4


3 or

5 x

4
3
9
 k
9
 k already in lowest terms

Recognize equivalent forms of rational expressions.
When working with rational expressions, it is important to be able to recognize equivalent forms of an expressions. For example, the common fraction can also be written and Consider the rational expression
5
6

6
5

6

2 x

3
.
2
The − sign representing the factor −1 is in front of the expression, even with fraction bar. The factor −1 may instead be placed in the numerator or in the denominator. Some other equivalent forms of this rational expression are
 
2 x

3

2 and
2 x

3

2

Recognize equivalent forms of rational expressions.
(cont’d)
By the distributive property,
 
2 x

3

2 can also be written
 
3
.
2

2 x

3
2
 
2 x

3
 is not an equivalent form of . The sign preceding 3 in
2 the numerator of should be − rather than +.
2
Be careful to apply the distributive property correctly.

EXAMPLE 7
Writing Equivalent Forms of a Rational Expression
Write four equivalent forms of the rational expression.

2 x

7 x

3
Solution:
 
2 x

7
 x

3
,
 
7
, x

3
2 x

7

3
 ,
2 x

7
3

7.2
Multiplying and Dividing Rational Expressions
1
2
Multiply rational expressions.
Divide rational expressions.

Multiply rational expressions.
The product of two fractions is found by multiplying the numerators and multiplying the denominators. Rational expressions are multiplied in the same way.
Multiplying Rational Expressions
P
The product of the rational expressions and is
Q
R
S
P R

PR
Q S QS
That is, to multiply rational expressions, multiply the numerators and multiply the denominators.

EXAMPLE 1
Multiplying Rational Expressions
Multiply. Write each answer in lowest terms.
Solution:
7 10
 
7

1
7
8
2 p q

3
9 pq
2

8
 p
2  
9
3
2

8 p
3 p q p q q
3

24 p q
It is also possible to divide out common factors in the numerator and denominator before multiplying the rational expressions.

EXAMPLE 2
Multiplying Rational Expressions
Multiply. Write the answer in lowest terms.
3
 p
 q

 p 2
 q p
 q

Solution:

2
3
 
 p q

 q
 p q p

2
3



 p
 q p
 q



 q p

3 q
2 p

EXAMPLE 3
Multiplying Rational Expressions
Multiply. Write the answer in lowest terms.
x
2 
3 x
7 x


6
10
 x
2
6
 x
2
 x
6

15
Solution:

 x
2 
7

3 x

6
 x


10
 
6 x

6 x
2 
2 x

15



2
3
 x
 x


2
5

 x x


5

2

 x
 x


3

1



 x
3 x


2
6
 
 x
 x


5
5
 
 
6 x

6 x

3



2 x

1

  x

3


Divide rational expressions.
P
Q
R
S
Dividing Rational Expressions
R
P R P S PS
Q S Q R QR
.
That is, to divide one rational expression by another rational expression, multiply the first rational expression by the reciprocal of the second rational expression.

EXAMPLE 4
Dividing Rational Expressions
Divide. Write each answer in lowest terms.
y

2 x

9.
Solution:
3

5
4 16 4 5

3
 
4
4

5

12
5
9 p
2
6 p
3
3 p

4

3 p

4

9 p
2
3 p

4

3 p

4
6 p
3

2
3

3 p
  p

3
2 p

EXAMPLE 5
Dividing Rational Expressions
Divide. Write the answer in lowest terms.
5
2 a b

10 ab
2 8
2
Solution:

5
2 a b

8
2 10 ab
2

5 a
2 b b

2 a b

EXAMPLE 6
Dividing Rational Expressions
Divide. Write the answer in lowest terms.
4
2
 x


1
3


 x
2
 x

4 x
2 
1
3

Solution:


4
  x
2
 x x


1
3




1
4 x
2
2 x x


1

3

4
 x

2 x

1

 

2 x

1 x


2 x

1
 x

4 2 x

1
 x

EXAMPLE 7
Dividing Rational Expressions (Factors Are Opposites)
Divide. Write in the answer in lowest terms.
a
2

1
2
 a
2

2 a

1
Solution:
 a
2


1
2
 a
2 
2 a

1

  a
 
1
 a

 a b a
1
  a


1


1 a


 a b


1
 a a
 a


1
1


Remember to write −1 when dividing out factors that are opposite of each other. It may be written in the numerator or denominator, but not both.

Multiplying or Dividing Rational Expressions.
Multiplying or Dividing Rational Expressions
Step 1: Note the operation.
If the operation is division, use the definition of division to rewrite it as multiplication.
Step 2: Factor all numerators and denominators completely.
Step 3: Multiply numerators and denominators.
Step 4: Write in lowest terms using the fundamental property.

7.3
Least Common Denominators
1 Find the least common denominator for a group of fractions.
2 Write equivalent rational expressions.

Find the least common denominator for a group of fractions.
Adding or subtracting rational expressions often requires a least common denominator (LCD), the simplest expression that is divisible by all of the denominators in all of the expressions. For example, the
2 5 least common denominator for the fractions and is 36 , because
9 12
36 is the smallest positive number divisible by both 9 and 12 .
We can often find least common denominators by inspection. For
1 2 example, the LCD for and is 6 m . In other cases, we find the
6 3 m
LCD by a procedure similar to that used for finding the greatest common factor.

Find the least common denominator for a group of fractions. (cont’d)
Finding the Least Common Denominator (LCD)
Step 1: Factor each denominator into prime factors.
Step 2: List each different denominator factor the greatest number of times it appears in any of the denominators.
Step 3: Multiply the denominator factors from Step 2 to get the
LCD.
When each denominator is factored into prime factors, every prime factor must be a factor of the least common denominator.

EXAMPLE 1
Finding the LCD
Find the LCD for each pair of fractions.
7 1
,
10 25
Solution:
10 2 5
25 5 5
LCD

5
2 
2
 

5
2

50
5
4 11
, m
4
8 12 m
6
8 m 4
12 m
6
LC D

2 3 3 m 6 m 4 m
6


2 3
2
 m
 
4
2 3 m
6

24 m 6

EXAMPLE 2
Finding the LCD
Find the LCD for
4
16
3 m n
5
and .
5
9 m
Solution:
16
3 m n 2 2 2 2
3 
9 m
5 m
5
LCD

4
 
5


2
4  m
3  n
  m
5

5
144 m n
When finding the LCD, use each factor the greatest number of times it appears in any single denominator, not the total number of times it appears.

EXAMPLE 3
Finding LCDs
Find the LCD for the fractions in each list.
Solution:
2
6

4
, x x x
3 x

1
2 
16
 x

4

 x


4
4

   
4

  x

4
  x

4

LCD
  
4
 x

4
 x
4 1

,
1 1
 x
Either x − 1 or 1 − x, since they are opposite expressions.

Write equivalent rational expressions.
Writing A Rational Expression with a Specified
Denominator
Step 1: Factor both denominators.
Step 2: Decide what factor (s) the denominator must be multiplied by in order to equal the specified denominator.
Step 3: Multiply the rational expression by the factor divided by itself.
(That is, multiply by 1.)

EXAMPLE 4
Writing Equivalent Rational Expressions
Rewrite each rational expression with the indicated denominator.
3
4

?
36
Solution:
3
4

4
?

9
3
4 4
9
9

27
36
7 k

?
5 30 k
7 k
5

?
5

6 k
7 k

7
5 5 k

6
6 k k

42 k
2
30 k

EXAMPLE 5
Writing Equivalent Rational Expressions
Rewrite each rational expression with the indicated denominator.
Solution:
2 a
9

5

6 a
?

15 k
5
2 k


2
1 k

 
2
?
 k

1

2 a
9

5

3

2 a
?

5

2 a
9

5


2 a
9

5


3
3

5 k

1 k k

2


 
2
?
  k

1


27
6 a

15

5 k


1
2



5 k


1
2



 k k


1
1





5 k


1
2

 k k


1
1



7.4
Adding and Subtracting Rational Expressions
1 Add rational expressions having the same denominator.
2 Add rational expressions having different denominators.
3 Subtract rational expressions.

Add rational expressions having the same denominator.
We find the sum of two rational expressions with the same procedure used for adding two fractions having the same denominator.
Adding Rational Expressions (Same Denominator)
If and ( Q ≠ 0) are rational expressions, then
Q
R
Q
P R P

R
 
Q Q Q
.
That is, to add rational expressions with the same denominator, add the numerators and keep the same denominator.

EXAMPLE 1
Adding Rational Expressions (Same Denominator)
Add. Write each answer in lowest terms.
Solution:
7

3
15 15
2
 x

2
 y x y x y

15

10
15

2

5
3

5

2 x

2 y x
 y

2
 x
 y
 x
 y

2

2
3

Add rational expressions having different denominators.
We use the following steps to add fractions having different denominators.
Adding Rational Expressions (Different Denominators)
Step 1: Find the least common denominator (LCD).
Step 2: Rewrite each rational expression as an equivalent rational expression with the LCD as the denominator.
Step 3: Add the numerators to get the numerator of the sum. The LCD is the denominator of the sum.
Step 4: Write in lowest terms using the fundamental property.

EXAMPLE 2
Adding Rational Expressions (Different Denominators)
Add. Write each answer in lowest terms.
Solution:
1

1
10 15

1
1 0

3
3

1
15

2
2
10 2
 
5
15 3 5
LCD 2 3 5 30

3

2
30 30

30

5
30

1
6 m

2
3 n 7 n
 m
3 n

7
7

2
7 n

3
3

7 m

6
21 n 21 n

7 m

6
21 n
LCD
   
21 n

EXAMPLE 3
Adding Rational Expressions
Add. Write the answer in lowest terms.
p

2

1
 p
4
2 p

1
Solution:
 p

2

1

 p

1
4
 p p

1

 p

2

1
 p p

1

1

 p

1
4
 p p

1





2 p

2

4 p

1
 p


1
 p

2 p

2
 p

1
 p

1


 p
2


1 p
 
 p
1


1


2 p

1

EXAMPLE 4
Adding Rational Expressions
Add. Write the answer in lowest terms.
2 k k
2 
5 k

4
 k
2
3

1
Solution:

 k

4
2 k
 k

1
  k

1
3
 k

1


 k

4
2
 k k

1



 k k


1
1



 k

1
3
 k

1



 k k


4
4



 k

4
2
 k
 
1



1 k

1


 k

4
3

 k k


1
4

 k

1


2 k
2 
5 k

12
 k

4
 k

1
 k

1
 
 k


2 k

3
4
 k
 k


4


1 k

1


2 k
2 
2 k

3 k

12
 k

4
 k

1
 k

1


EXAMPLE 5
Adding Rational Expressions (Denominators Are Opposites)
Add. Write the answer in lowest terms.
2 m m

3 n

3 n n

2 m

Solution:
2 m m

3 n

3 n n

2 m

 
 

2 m m

3 n

2 m

 n
3 n

2 m m

3 n

 n
 
2 m

2 m

3 n
or
3 n

2 m

Subtract rational expressions.
Subtracting Rational Expressions (Same Denominator)
R
Q ≠ 0) are rational expressions, then
Q Q
P

R

P

Q Q Q
R
That is, to subtract rational expressions with the same denominator, subtract the numerators and keep the same denominator.
We subtract rational expressions having different denominators using a procedure similar to the one used to add rational expressions having different denominators.

EXAMPLE 6
Subtracting Rational Expressions (Same Denominator)
Subtract. Write the answer in lowest terms.
t
5
 t
1
 t
5


1 t
Solution:

5 t
 t
 t

1

5 t
  t t

1

4 t

5 t

1
Sign errors often occur in subtraction problems. The numerator of the fraction being subtracted must be treated as a single quantity. Be sure to use parentheses after the subtraction sign.

EXAMPLE 7
Subtracting Rational Expressions (Different Denominators)
Subtract. Write the answer in lowest terms.
a
6

2
 a
1

3
Solution:
 a
6

2
 a a


3
3
 a
1

3
 a

2 a

2


6 a

18 a
 a

2
 
 a

3

2


 a

6 a

18
2
 a

3
  a


2 a

3 a

2



6 a 18 2 a

2
 a

3



 a

5 a

20
2
 a

3


5
 a


4
 a

2 a

3


EXAMPLE 8
Subtracting Rational Expressions (Denominators Are Opposites)
Subtract. Write the answer in lowest terms. x
4
 x

 
1 1
 x
1
Solution:
 x
4
 x

 
1 1
 x
1

 
 
 x
4
 x
1

3 x x


1
1

4 x
 
3 x

1
 x

1

4 x

3 x

1 x

1
 x

1 x

1

1

EXAMPLE 9
Subtracting Rational Expressions
Subtract. Write the answer in lowest terms.
3
 r
 r
2
5 r r
2
4

10 r

25
Solution:


3 r

5
  r

5
4
 r

5



3 r
2 
15 r

5
 r

5
 r r

5
4 r
 r

5

 r
 r r

3 r

19



5 r

5


3 r
 
5

 r r


5
5

 r

5
4
 r

5

 r r

3 r
2 
19 r
 
5
 r

5



3 r

19

 r

5

2

7.5
Complex Fractions
1
Simplify a complex fraction by writing it as a division problem
(Method 1).
2 Simplify a complex fraction by multiplying numerator and denominator by the least common denominator (Method 2).

Complex Fractions.
1

1
The quotient of two mixed numbers in arithmetic, such as can
2 4 be written as a fraction.
2
1

1
2
1

3
1
2 4
 2
1

2

2
1
3
4
3
4
In algebra, some rational expressions also have fractions in the numerator, or denominator, or both.
Complex Fraction
A quotient with one or more fractions in the numerator, or denominator, or both is called a complex fraction.
The parts of a complex fraction are named as follows.
2

1
2
1
3

4
Numerator of complex fraction
Main fraction bar
Denominator of complex fraction

Simplify a complex fraction by writing it as a division problem (Method 1).
Since the main fraction bar represents division in a complex fraction, one method of simplifying a complex fraction involves division.
Method 1 for Simplifying a Complex Fraction
Step 1: Write both the numerator and denominator as single fractions.
Step 2: Change the complex fraction to a division problem.
Step 3: Perform the indicated division.

EXAMPLE 1
Simplifying Complex Fractions (Method 1)
Simplify each complex fraction.
Solution
:
2

1
5 4
1

1
2 3
6 m m

1

2
3
4 m
2
 5
1
2
4
3
3
1
4
3


1 m

2 m
2
1

4 m
5
5
2
2

8

5
20 20
3

2
6 6
13
 20
5
6


2 m

1
2 2 m

1


2 m

1

2 m

1


4 m
2 m
2

1

3

2 m
2 m

1

4 m

13

5
20 6

13 6
20 5

2

39
2

5 0

78
100

39
50

2 m

1

2

2 m
3
 m
4 m

1


EXAMPLE 2
Simplifying a Complex Fraction (Method 1)
Simplify the complex fraction.
2 3 m n p
4 m n p
2
Solution:


2 3 m n
 p p
4 m n
2
    n p

2 3 m n
 p
2
4 p m n p
 p
 m
    n
 m
2 p n
3
 p m
4
2 n

2 n p m
2

EXAMPLE 3
Simplifying a Complex Fraction (Method 1)
Simplify the complex fraction.
1 a

1
2

 b

2
2 b

1
2 a

3
Solution
:




 b b a a




2
2
3
3





 b a
1

1
2

2

 b
2

2
 a
1

3




 a a b b




1
1
2
2





 b

  a

 
1
 b
2 a


2

2


2 a 6
 b

2
 
 a

3

2


 a

2

1
 b

2
  b
2

2
 a

8
3

 a

2

1
 b

2


 b

2
  a

3

2
 
8

 a

2
 a
 
 
3
 

8


Simplify a complex fraction by multiplying numerator and denominator by the least common denominator
(Method 2).
Since any expression can be multiplied by a form of 1 to get an equivalent expression, we can multiply both the numerator and denominator of a complex fraction by the same nonzero expression to get an equivalent rational expression. If we choose the expression to be the LCD of all the fractions within the complex fraction, the complex fraction will be simplified.
Method 2 for Simplifying a Complex Fraction
Step 1: Find the LCD of all fractions within the complex fraction.
Step 2: Multiply both the numerator and denominator of the complex fraction by this LCD using the distributive property as necessary. Write in lowest terms.

EXAMPLE 4
Simplifying Complex Fractions (Method 2)
Simplify each complex fraction.
Solution:
2

1
3 4
4

1
9 2

3 6
3 6
2

1
3 4
4
9

1
2
 
15
34
2

6 a
4
3
 a
 a a 





2
3


6 a
4 a







2 a

6
3 a

4

EXAMPLE 5
Simplifying a Complex Fraction (Method 2)
Simplify the complex fraction.
2

3
2 a b ab
2
4

1
2 2 a b ab
Solution:

2 2 a b
2 2 a b
2

3
2 a b ab
2
4

1
2 2 a b ab

2 b

3 a
4
 ab

EXAMPLE 6
Deciding on a Method and Simplifying Complex Fractions
Simplify each complex fraction.
Solution:
1

2 x x

1 x
4

1

 
1
 x
 x

1

1
 x x
2

1 x
4

1

 x 1

2 x
4 x

3 x

1
4 x
2 x

3 x


4 x
2
4 9 x
2 
16

 x

4
 x

 x
4


4
 x




4


2 x

3 2 x

3 x




2 x

3
4 x


4



   x

4





 x

4
2 x

3
 
2 x

3
2 x

3

   

 x

4
2 x

3

 
Remember the same answer is obtained regardless of whether Method 1 or
Method 2 is used. Some students prefer one method over the other.