Sixth Edition
James Stewart  Lothar Redlin  Saleem Watson

P

P.7

Fractional Expression
A quotient of two algebraic expressions is called a fractional expression .
• Here are some examples: x
2 x y

1 y
2

2

4 x
3  x x
2 
5 x

6 x x 2 
1

Rational Expression
A rational expression is a fractional expression in which both the numerator and denominator are polynomials.
• Here are some examples: x
2 x

1 y y
2

2

4 x
3  x x
2 
5 x

6

Rational Expressions
• How to perform algebraic operations on rational expressions.

The Domain of an Algebraic Expression
In general, an algebraic expression may not be defined for all values of the variable.
The domain of an algebraic expression is:
• The set of real numbers that the variable is permitted to have.

The Domain of an Algebraic Expression
The table gives some basic expressions and their domains.

E.g. 1 —Finding the Domain of an Expression
Find the domains of these expressions.
  x
2 
3 x

1 x x
2 
5 x

6 x x

5

E.g. 1 —Finding the Domain
2 x 2 + 3 x – 1
Example (a)
This polynomial is defined for every x .
• Thus, the domain is the set of real numbers.

E.g. 1 —Finding the Domain
Example (b)
We first factor the denominator: x x 2 
5 x

6

 x

2 x
 x

3

• Since the denominator is zero when x = 2 or x = 3.
• The expression is not defined for these numbers.
• The domain is: { x | x ≠ 2 and x ≠ 3}.

E.g. 1 —Finding the Domain
Example (c) x x

5
• For the numerator to be defined, we must have x ≥ 0.
• Also, we cannot divide by zero, so x ≠ 5.
• Thus the domain is { x | x ≥ 0 and x ≠ 5}.

Simplifying Rational Expressions
To simplify rational expressions , we factor both numerator and denominator and use following property of fractions:
AC

A
BC B
• This allows us to cancel common factors from the numerator and denominator.

E.g. 2 —Simplifying Rational Expressions by Cancellation
x 2 
1 x 2  
2 x 2 
1
 x 2  

 x x


1
1


2 x x


1
2


 x x

1

2
(Factor )
(Cancel common factors)

Caution
We can’t cancel the x 2 ’s in x
2 
1 x
2  
2 because x 2 is not a factor.

Multiplying Rational Expressions
To multiply rational expressions , we use the following property of fractions:
A C

AC
B D BD
This says that:
• To multiply two fractions, we multiply their numerators and multiply their denominators.

E.g. 3 —Multiplying Rational Expressions
Perform the indicated multiplication, and simplify: x x
2
2


8
2 x x


3
16

3 x x


12
1

E.g. 3 —Multiplying Rational Expressions
We first factor: x x
2
2


8
2 x x


3
16

3 x x


12
1

 x

 x
1

 x
4


2
3


3
 x x


1
4



3
 x

1
 x

3
 x

4

 x

1
 x

4

2
3
 x

3
 x

4
Factor
Property of fractions
Cancel common factors

Dividing Rational Expressions
To divide rational expressions , we use the following property of fractions:
A

C

A D
B D B C
This says that:
• To divide a fraction by another fraction, we invert the divisor and multiply.

E.g. 4 —Dividing Rational Expressions
Perform the indicated division, and simplify: x x
2

4

4
 x
2 
3 x

4 x
2 
5 x

6

E.g. 4 —Dividing Rational Expressions x x
2


4
4


 x
2 
3 x

4 x
2 
5 x

6
 x x x
2




4 x
4
2


 x 2 
5 x

6 x
2 
3 x

4
4 x

 x
2


2 x

 x
4


3 x


1


 x
 x
2


3 x

1

Invert diviser and multiply
Factor
Cancel common factors

Adding and Subtracting Rational Expressions
To add or subtract rational expressions , we first find a common denominator and then use the following property of fractions:
A

B

A

C C C
B

Adding and Subtracting Rational Expressions
Although any common denominator will work, it is best to use the least common denominator (LCD) as explained in Section
P.2.
• The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.

Caution
Avoid making the following error:
A

A

A
B C

Caution
For instance, if we let A = 2, B = 1, and C = 1, then we see the error:
2

?
2 2
1
2
2
?
2 2
?
1 4

E.g. 5 —Adding and Subtracting Rational Expressions
Perform the indicated operations, and simplify:
(a) x
3

1
 x x

2
(b) x
2
1

1

2
 x

1

2

E.g. 5 —Adding Rational Exp.
Example (a)
Here LCD is simply the product ( x – 1)( x + 2).
 x
3

 x
1

3 x

2

1 x
 x

  x
2

2


 x x x


1

 x
1


2


3 x 6 x 2  x
 x

1
 x

2

 x 2 
2 x

6
 x

1
 x

2

Add fractions
Fractions by LCD
Combine terms in numerator

E.g. 5 —Subtracting Rational Exp.
Example (b)
The LCD of x 2 – 1 = ( x – 1)( x + 1) and ( x + 1) 2 is ( x – 1)( x + 1) 2 .
x 2
1

1

2
 x

1

2


 x

1
1
 x

1
  x
2

1

2
 x
  x

1

 x

1
 x

1

2
Factor
Combine fractions using LCD

E.g. 5 —Subtracting Rational Exp.
 x
  x

2
 x

1
 x

1

2
Example (b)
Distributive Property

3
 x
 x

1
 x

1

2
Combine terms in numerator

Compound Fraction
A
is:
• A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.

E.g. 6 —Simplifying a Compound Fraction
x

1 y
1
 y x

E.g. 6 —Simplifying
Solution 1
1. We combine the terms in the numerator into a single fraction.
2. We do the same in the denominator.
3. Then we invert and multiply.

E.g. 6 —Simplifying
Thus,
Solution 1
1 x y


1 x
 y y
 x y
 y x x
 x
 y y
 x x
 y



 y y x
 y



E.g. 6 —Simplifying
Solution 2
1. We find the LCD of all the fractions in the expression,
2. Then multiply the numerator and denominator by it.

E.g. 6 —Simplifying
Solution 2
Here, the LCD of all the fractions is xy .
x y
1


1 y x


 x

1 y

1
 y x x
2  xy xy xy xy
 y 2


 y y x
 y


Multiply num. and denom. by xy
Simplify
Factor

Simplifying a Compound Fraction
The next two examples show situations in calculus that require the ability to work with fractional expressions.

E.g. 7 —Simplifying a Compound Fraction
1

1 a
 h a h
• We begin by combining the fractions in the numerator using a common denominator:

E.g. 7 —Simplifying a Compound Fraction

 a
1
 h

1 a a h
  a
 a a
 h h
 

Combine fractions in numerator a


 h a

 h



1 a a h h
Invert divisor and multiply

E.g. 7 —Simplifying a Compound Fraction


 a

 
 h


1 a a h h

 h a a
 h


1 h


1 a a
 h

DistributiveProperty
Simplify
Cancel common factors

E.g. 8 —Simplifying a Compound Fraction

1
 x
2
 1
2  x
2

1
 x
2
  1
2
1
 x 2

E.g. 8 —Simplifying
Solution 1
Factor (1 + x 2 )
–1/2 from the numerator.

1
 x 2

1/2
1

 x x
2
2

1
 x 2
 
1/2



1
 x
2

1/2

1
 x
2
 
1/2
1


1 x
 x
2  x
2

1
 x 2
2

1

1
 x
2

3/2

E.g. 8 —Simplifying
Solution 2
Since (1 + x 2 )
–1/2
= 1/(1 + x 2 ) 1/2 is a fraction, we can clear all fractions by multiplying numerator and denominator by (1 + x 2 ) 1/2 .

E.g. 8 —Simplifying
Solution 2
Thus,



1
 x 2

1/2  x 2

1
 x 2
 
1/2

1
 x 2

1
1/2

1
 x
 x
2 x
2
2

1
 x 2
 

1
1

 x x
2
2


1/2
1/2

1

 x 2  x
1
 x 2


3/2
2


1

1 x 2

3/2

Rationalizing the Denominator
If a fraction has a denominator of the form we may rationalize the denominator by multiplying numerator and denominator by the


Rationalizing the Denominator
This is effective because, by Product Formula
1 in Section P.5, the product of the denominator and its conjugate radical does not contain a radical:

A B C




A 2  2

E.g. 9 —Rationalizing the Denominator
1
1

2
• We multiply both the numerator and the denominator by the conjugate radical of 1

2

E.g. 9 —Rationalizing the Denominator
Thus,
1

1
2

1

1
2

1

2
1

2

1
2
1

2

 
2
Product Formula 1

1

2

1

2

1
 

E.g. 10 —Rationalizing the Numerator
4 h
2
• We multiply numerator and denominator

E.g. 10 —Rationalizing the Numerator
Thus,
4 2 h


4 h h


4

4 h

2
2


2 2
2

4
4
2
2
Product Formula 1

E.g. 10 —Rationalizing the Numerator
 h

4
4
 h

4
 
4
1 h
2
4
2


2

Cancel common factors

Avoiding Common Errors
Don’t make the mistake of applying properties of multiplication to the operation of addition.
• Many of the common errors in algebra involve doing just that.

Avoiding Common Errors
The following table states several multiplication properties and illustrates the error in applying them to addition.

Avoiding Common Errors
To verify that the equations in the right-hand column are wrong, simply substitute numbers for a and b and calculate each side.

Avoiding Common Errors
For example, if we take a = 2 and b = 2 in the fourth error, we get different values for the left- and right-hand sides:

Avoiding Common Errors
The left-hand side is:
1
    a b 2 2
1
The right-hand side is:
1 a
 b

2
1

2

1
4
• Since 1 ≠ ¼, the stated equation is wrong.

Avoiding Common Errors
You should similarly convince yourself of the error in each of the other equations.
• See Exercise 119.