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College AlgebraSixth EditionJames Stewart Lothar Redlin Saleem Watson
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PrerequisitesP
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Rational ExpressionsP.7
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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 25 6 1
x x x
x x x
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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
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Rational Expressions
In this section, we learn:
• How to perform algebraic operations on rational expressions.
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The Domain of
an Algebraic Expression
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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
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The Domain of an Algebraic Expression
The table gives some basic expressions
and their domains.
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E.g. 1—Finding the Domain of an Expression
Find the domains of these expressions.
2
2
a 2 3 1
b5 6
c5
x x
x
x x
x
x
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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)
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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
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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
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Simplifying
Rational Expressions
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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
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E.g. 2—Simplifying Rational Expressions by Cancellation
Simplify:
2
2
1
2
x
x x
2
2
)(Factor
(Cancel common factors)
1
1
21
2
2
1
1
x
x xxx
x
x
x
x
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Caution
We can’t cancel the x2’s in
because x2 is not a factor.
2
2
1
2
x
x x
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Multiplying and Dividing
Rational Expressions
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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
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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
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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 11 3 3 4
14
3 3
3 3
41
41
4
x
x x x
x x xx x x
xx
x
x
x
x
x x
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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
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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
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E.g. 4—Dividing Rational Expressions
2
2 2
2
2 2Invert diviser and multiply
Factor
Cancel common factors
4 3 4
4 5 6
4 5 6
4 3 43
2 1
3
1
4
2 4
2
2
x x x
x x x
x x x
x x xx
x x
x
x
x
x
x
x
x
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Adding and Subtracting
Rational Expressions
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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
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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.
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Caution
Avoid making the following error:
A
B C
A A
B C
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Caution
For instance, if we let A = 2, B = 1, and C = 1,
then we see the error:
?
?
?
2 2 2
1 1 1 12
2 22
1 4 Wrong!
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E.g. 5—Adding and Subtracting Rational Expressions
Perform the indicated operations, and
simplify:
22
3(a)
1 2
1 2(b)
1 1
x
x x
x x
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E.g. 5—Adding Rational Exp.
2
2
Fractions by LCD
Add fractions
Combine terms in numerator
3
1 23 2 1
1 2 1 2
3 6
1 2
2 6
1 2
x
x xx 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).
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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)
22
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
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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)
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Compound Fractions
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Compound Fraction
A compound fraction is:
• A fraction in which the numerator, the denominator, or both, are themselves fractional expressions.
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E.g. 6—Simplifying a Compound Fraction
Simplify:
1
1
xy
yx
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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
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E.g. 6—Simplifying
Thus,
1
1
x x yy y
y x yx x
x y x
y x y
x x y
y x y
Solution 1
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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
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E.g. 6—Simplifying
Here, the LCD of all the fractions is xy.
2
2
Multiply num.and denom.by
Simplify
Factor
1 1
1 1xy
x xy y
y yx x
x xy
xy y
x x y
y x y
xy
xy
Solution 2
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Simplifying a Compound Fraction
The next two examples show situations
in calculus that require the ability to work
with fractional expressions.
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E.g. 7—Simplifying a Compound Fraction
Simplify:
• We begin by combining the fractions in the numerator using a common denominator:
1 1a h a
h
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E.g. 7—Simplifying a Compound Fraction
Combine fractions in numerator
Invert divisor and multiply
1 1
1
a h ah
a a h
a a h
ha a h
a a h h
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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
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E.g. 8—Simplifying a Compound Fraction
Simplify:
1 1
2 2 22 2
2
1 1
1
x x x
x
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E.g. 8—Simplifying
Factor (1 + x2)–1/2 from the numerator.
1/21/2 1/2 2 2 22 2 2
2 2
1/22
2
3/22
1 11 1
1 1
1
11
1
x x xx x x
x x
x
x
x
Solution 1
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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
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E.g. 8—Simplifying
1/2 1/22 2 2
2
1/2 1/2 1/22 2 2 2
1/22 2
2 2
3/2 3/22 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,
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Rationalizing the Denominator
or the Numerator
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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
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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 2A B C A B C A B C
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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
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Thus,
E.g. 9—Rationalizing the Denominator
22
Product Formula1
1 1
1 2 1 2
1 2
1 2
1 2
1 2
1 22 1
1 2 1
1 2
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E.g. 10—Rationalizing the Numerator
Rationalize the numerator:
• We multiply numerator and denominator by the conjugate radical :
4 2h
h
4 2h
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E.g. 10—Rationalizing the Numerator
Thus,
22
Product Formula1
4 2
4 2
4 2
4 2
4
4 2
2
h
h
h
h
h
h
h
h h
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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
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Avoiding Common Errors
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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.
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Avoiding Common Errors
The following table states several
multiplication properties and illustrates the
error in applying them to addition.
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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.
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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:
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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
12 2a b
1 1 1
2 2 4a b
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Avoiding Common Errors
You should similarly convince yourself
of the error in each of the other equations.• See Exercise 119.