4. Fractional Expression
A quotient of two algebraic expressions
is called a fractional expression.
• Here are some examples:
2
2 2
1 4
x y
x y
3
2 2
5 6 1
x x x
x x x
5. Rational Expression
A rational expression is a fractional
expression in which both the numerator
and denominator are polynomials.
• Here are some examples:
3
2 2
2 2
1 4 5 6
x y x x
x y x x
8. 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
9. The Domain of an Algebraic Expression
The table gives some basic expressions
and their domains.
10. E.g. 1—Finding the Domain of an Expression
Find the domains of these expressions.
2
2
a 2 3 1
b
5 6
c
5
x x
x
x x
x
x
11. E.g. 1—Finding the Domain
2x2 + 3x – 1
This polynomial is defined for every x.
• Thus, the domain is the set
of real numbers.
Example (a)
12. E.g. 1—Finding the Domain
We first factor the denominator:
• 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}.
Example (b)
2
5 6 2 3
x x
x x x x
13. E.g. 1—Finding the Domain
• 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}.
Example (c)
5
x
x
15. Simplifying Rational Expressions
To simplify rational expressions, we
factor both numerator and denominator
and use following property of fractions:
• This allows us to cancel common factors
from the numerator and denominator.
AC A
BC B
16. E.g. 2—Simplifying Rational Expressions by Cancellation
Simplify:
2
2
1
2
x
x x
2
2
)
(Factor
(Cancel common factors)
1
1
2
1
2
2
1
1
x
x x
x
x
x
x
x
x
19. Multiplying Rational Expressions
To multiply rational expressions, we use
the following property of fractions:
This says that:
• To multiply two fractions, we multiply their
numerators and multiply their denominators.
A C AC
B D BD
20. E.g. 3—Multiplying Rational Expressions
Perform the indicated multiplication,
and simplify:
2
2
2 3 3 12
8 16 1
x x x
x x x
21. E.g. 3—Multiplying Rational Expressions
We first factor:
2
2
2
2
Factor
Property of fractions
Cancel common factors
2 3 3 12
8 16 1
1 3 3 4
1
4
3 3
3 3
4
1
4
1
4
x
x x x
x x x
x x x
x
x
x
x
x
x
x x
22. Dividing Rational Expressions
To divide rational expressions, we use
the following property of fractions:
This says that:
• To divide a fraction by another fraction,
we invert the divisor and multiply.
A C A D
B D B C
23. E.g. 4—Dividing Rational Expressions
Perform the indicated division, and
simplify:
2
2 2
4 3 4
4 5 6
x x x
x x x
24. E.g. 4—Dividing Rational Expressions
2
2 2
2
2 2
Invert diviser and multiply
Factor
Cancel common factors
4 3 4
4 5 6
4 5 6
4 3 4
3
2 1
3
1
4
2 4
2
2
x x x
x x x
x x x
x x x
x
x x
x
x
x
x
x
x
x
26. 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 B
C C C
27. 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.
29. Caution
For instance, if we let A = 2, B = 1, and C = 1,
then we see the error:
?
?
?
2 2 2
1 1 1 1
2
2 2
2
1 4 Wrong!
30. E.g. 5—Adding and Subtracting Rational Expressions
Perform the indicated operations, and
simplify:
2
2
3
(a)
1 2
1 2
(b)
1 1
x
x x
x x
31. E.g. 5—Adding Rational Exp.
2
2
Fractions by LCD
Add fractions
Combine terms in numerator
3
1 2
3 2 1
1 2 1 2
3 6
1 2
2 6
1 2
x
x x
x x x
x x x x
x x x
x x
x x
x x
Example (a)
Here LCD is simply the product (x – 1)(x + 2).
32. E.g. 5—Subtracting Rational Exp.
The LCD of x2 – 1 = (x – 1)(x + 1) and (x + 1)2
is (x – 1)(x + 1)2.
Example (b)
2
2
2
2
Factor
Combine fractions using LCD
1 2
1 1
1 2
1 1 1
1 2 1
1 1
x x
x x x
x x
x x
33. E.g. 5—Subtracting Rational Exp.
2
2
Distributive Property
Combine terms in numerator
1 2 2
1 1
3
1 1
x x
x x
x
x x
Example (b)
37. E.g. 6—Simplifying
One solution is as follows.
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.
Solution 1
39. E.g. 6—Simplifying
Another solution is as follows.
1. We find the LCD of all the fractions in
the expression,
2. Then multiply the numerator and denominator
by it.
Solution 2
40. E.g. 6—Simplifying
Here, the LCD of all the fractions is xy.
2
2
Multiply num.anddenom.by
Simplify
Factor
1 1
1 1
xy
x x
y y
y y
x x
x xy
xy y
x x y
y x y
xy
xy
Solution 2
41. Simplifying a Compound Fraction
The next two examples show situations
in calculus that require the ability to work
with fractional expressions.
42. E.g. 7—Simplifying a Compound Fraction
Simplify:
• We begin by combining the fractions in
the numerator using a common denominator:
1 1
a h a
h
43. E.g. 7—Simplifying a Compound Fraction
Combine fractions in numerator
Invert divisor and multiply
1 1
1
a h a
h
a a h
a a h
h
a a h
a a h h
44. E.g. 7—Simplifying a Compound Fraction
DistributiveProperty
Simplify
Cancel common factors
1
1
1
a a h
a a h h
a a h
a a h
h
h
45. E.g. 8—Simplifying a Compound Fraction
Simplify:
1 1
2 2 2
2 2
2
1 1
1
x x x
x
46. E.g. 8—Simplifying
Factor (1 + x2)–1/2 from the numerator.
1/2
1/2 1/2 2 2 2
2 2 2
2 2
1/2
2
2
3/2
2
1 1
1 1
1 1
1
1
1
1
x x x
x x x
x x
x
x
x
Solution 1
47. E.g. 8—Simplifying
Since (1 + x2)–1/2 = 1/(1 + x2)1/2 is a fraction,
we can clear all fractions by multiplying
numerator and denominator by (1 + x2)1/2.
Solution 2
48. E.g. 8—Simplifying
1/2 1/2
2 2 2
2
1/2 1/2 1/2
2 2 2 2
1/2
2 2
2 2
3/2 3/2
2 2
1 1
1
1 1 1
1 1
1 1
1 1
x x x
x
x x x x
x x
x x
x x
Solution 2
Thus,
50. Rationalizing the Denominator
If a fraction has a denominator of the form
we may rationalize the denominator by
multiplying numerator and denominator
by the conjugate radical .
A B C
A B C
51. 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:
2 2
A B C A B C A B C
52. E.g. 9—Rationalizing the Denominator
Rationalize the denominator:
• We multiply both the numerator and
the denominator by the conjugate radical
of , which is .
1
1 2
1 2
1 2
54. E.g. 10—Rationalizing the Numerator
Rationalize the numerator:
• We multiply numerator and denominator
by the conjugate radical :
4 2
h
h
4 2
h
55. E.g. 10—Rationalizing the Numerator
Thus,
2
2
Product Formula1
4 2
4 2
4 2
4 2
4
4 2
2
h
h
h
h
h
h
h
h h
56. E.g. 10—Rationalizing the Numerator
Cancel common factors
4 4
4 2
4 2
1
4 2
h
h h
h
h h
h
58. 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.
59. Avoiding Common Errors
The following table states several
multiplication properties and illustrates the
error in applying them to addition.
60. 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.
61. 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:
62. Avoiding Common Errors
The left-hand side is:
The right-hand side is:
• Since 1 ≠ ¼, the stated equation is wrong.
1 1 1 1
1
2 2
a b
1 1 1
2 2 4
a b
63. Avoiding Common Errors
You should similarly convince yourself
of the error in each of the other equations.
• See Exercise 119.
