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SECTION 0.5
0.5
F R A C T I O N S A N D R AT I O N A L I Z AT I O N
■
■
■
Add and subtract rational expressions.
Simplify rational expressions involving radicals.
Rationalize numerators and denominators of rational expressions.
Operations with Fractions
In this section, you will review operations involving fractional expressions such as
2
,
x
x 2 2x 4
,
x6
and
1
x 2 1
.
The first two expressions have polynomials as both numerator and denominator
and are called rational expressions. A rational expression is proper if the degree
of the numerator is less than the degree of the denominator. For example,
x
x2 1
is proper. If the degree of the numerator is greater than or equal to the degree of
the denominator, then the rational expression is improper. For example,
x2
,
x2 1
x 3 2x 1
x1
and
are both improper.
Operations with Fractions
1. Add fractions (find a common denominator):
c
a d
c b
ad
bc
ad bc
a
,
b d b d
d b
bd bd
bd
b 0, d 0
2. Subtract fractions (find a common denominator):
a
c b
ad
c
a d
bc
ad bc
,
b d b d
d b
bd bd
bd
b 0, d 0
3. Multiply fractions:
abdc bdac ,
b 0, d 0
4. Divide fractions (invert and multiply):
dc adbc,
ab
a
cd
b
1c bca ,
a
ab ab
c
c1
b
b 0,
c 0, d 0
5. Divide out like factors:
ab b
,
ac
c
Fractions and Rationalization
ab ac ab c b c
,
ad
ad
d
a 0, c 0, d 0
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EXAMPLE 1
Adding and Subtracting Rational Expressions
Perform the indicated operation and simplify.
(a) x 1
x
(b)
1
2
x 1 2x 1
SOLUTION
(a) x (b)
1 x2 1
x
x
x
2
x 1
x
Write with common denominator.
Add fractions.
1
2
2x 1
2(x 1
x 1 2x 1 x 12x 1 x 12x 1
3
2x 1 2x 2
2
2x 2 x 1
2x x 1
TRY
IT
1
Perform each indicated operation and simplify.
(a) x 2
x
(b)
2
1
x 1 2x 1
In adding (or subtracting) fractions whose denominators have no common
factors, it is convenient to use the following pattern.
a
c
a
b d
b
c
d
ad bc
bd
For instance, in Example 1(b), you could have used this pattern as shown.
2
1
2x 1 2x 1
x 1 2x 1
x 12x 1
2x 1 2x 2
3
2
x 12x 1
2x x 1
In Example 1, the denominators of the rational expressions have no common
factors. When the denominators do have common factors, it is best to find the
least common denominator before adding or subtracting. For instance, when
adding 1x and 2x 2, you can recognize that the least common denominator is x 2
and write
x
2
1
2
2 2 2
x
x
x
x
x 2.
x2
Write with common denominator.
Add fractions.
This is further demonstrated in Example 2.
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SECTION 0.5
EXAMPLE 2
Adding and Subtracting Rational Expressions
Perform the indicated operation and simplify.
(a)
x
3
x2 1 x 1
(b)
1
1
2x 2 2x 4x
SOLUTION
(a) Because x 2 1 x 1x 1, the least common denominator is x 2 1.
x2
x
3
x
3
1 x 1 x 1x 1 x 1
x
3x 1
x 1x 1 x 1x 1
x 3x 3
x 1x 1
4x 3
2
x 1
Factor.
Write with
common
denominator.
Add fractions.
Simplify.
(b) In this case, the least common denominator is 4xx 2.
1
1
1
1
2x 2 2x 4x 2xx 2 22x
2
x2
22xx 2 22xx 2
2x2
4xx 2
x
4xx 2
1
, x0
4x 2
TRY
IT
Factor.
Write with
common
denominator.
Subtract
fractions.
Divide out
like factor.
Simplify.
2
Perform each indicated operation and simplify.
(a)
x
2
x 4 x2
2
ALGEBRA
(b)
1
1
3x 2x 3x
2
REVIEW
To add more than two fractions, you must find a denominator that is common to
all the fractions. For instance, to add 12, 13, and 15, use a (least) common denominator of 30 and write
1 1 1 15 10
6
2 3 5 30 30 30
31 .
30
Write with common denominator.
Add fractions.
Fractions and Rationalization
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A Precalculus Review
To add more than two rational expressions, use a similar procedure, as shown
in Example 3. (Expressions such as those shown in this example are used in calculus to perform an integration technique called integration by partial fractions.)
EXAMPLE 3
Adding More than Two Rational Expressions
Perform the indicated addition of rational expressions.
(a)
A
B
C
x2 x3 x4
(b)
C
A
B
x 2 x 22 x 1
SOLUTION
(a) The least common denominator is x 2x 3x 4.
B
C
A
x2 x3 x4
Ax 3x 4 Bx 2x 4 Cx 2x 3
x 2x 3x 4
Ax 2 x 12 Bx 2 6x 8 Cx 2 x 6
x 2x 3x 4
Ax 2 Bx 2 Cx 2 Ax 6Bx Cx 12A 8B 6C
x 2x 3x 4
A B Cx 2 A 6B C x 12A 8B 6C
x 2x 3x 4
(b) Here the least common denominator is x 2 2 x 1.
A
B
C
x 2 x 22 x 1
Ax 2x 1 Bx 1 Cx 2 2
x 2 2x 1
Ax 2 x 2 Bx 1 Cx 2 4x 4
x 2 2x 1
Ax 2 Cx 2 Ax Bx 4Cx 2A B 4C
x 2 2x 1
A Cx 2 A B 4Cx 2A B 4C
x 2 2x 1
TRY
IT
3
Perform each indicated addition of rational expressions.
(a)
A
B
C
x1 x1 x2
(b)
A
B
C
x 1 x 12 x 2
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SECTION 0.5
Expressions Involving Radicals
In calculus, the operation of differentiation tends to produce “messy” expressions
when applied to fractional expressions. This is especially true when the fractional
expression involves radicals. When differentiation is used, it is important to be
able to simplify these expressions so that you can obtain more manageable forms.
All of the expressions in Examples 4 and 5 are the results of differentiation. In
each case, note how much simpler the simplified form is than the original form.
EXAMPLE 4
Simplifying an Expression with Radicals
Simplify each expression.
x
2x 1
x1
x 1 (a)
(b)
x 1x
2
2x
1
1
2x 1 2
SOLUTION
x
2x 1
x
2x 1
2x 1 2x 1
x1
x1
2x 2 x
2x 1
x1
1
x2
1
2x 1 x 1
x2
2x 132
x 1 (a)
(b)
x 1x
2
To divide, invert and
multiply
Multiply.
2x
1
1
2 x 1
1
x
1
x x 1
x 1
2
x 1x
x 1x
IT
Subtract fractions.
2
TRY
Write with common
denominator.
2
x
1 x
2
2
2
1
x
1 x 2 1
x x 1
x 1 1
2
2
2
1
1
x 2
4
Simplify each expression.
x
x 2 4x 2
(a)
x2
(b)
x 1
x2 4
1 x
x2 4
Fractions and Rationalization
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EXAMPLE 5
Simplifying an Expression with Radicals
Simplify the expression.
x
2x2x 1 x
2
2
1
x2
x 1x
2
2x
1
1
2 x 1
2
SOLUTION From Example 4(b), you already know that the second part of this
sum simplifies to 1x 2 1. The first part simplifies as shown.
x
2x2x 1 x
2
2
x
1
x 2 1
x 2
x2
x x 1
x 2
x2 1
2 2
2 2
x x 1 x x 1
x 2 x 2 1
2 2
x x 1
1
2 2
x x 1
2
2 2
So, the sum is
x
2x2x 1 x
2
2
1
x2
x 1x
2x
1
1
2x 1 2
1
1
1
1
x2
2 2
x x 1 x 2x 2 1
x2 1
2 2
x x 1
x 2 1
.
x2
TRY
IT
1
2
x 2x 2
x 2
5
Simplify the expression.
x
3x3x 4 x
2
2
4
x2
ALGEBRA
x 1x
2
3x
1
4
3x 4 2
REVIEW
To check that the simplified expression in Example 5 is equivalent to the original expression, try substituting values of x into each expression. For instance,
when you substitute x 1 into each expression, you obtain 2.
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SECTION 0.5
Fractions and Rationalization
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Rationalization Techniques
In working with quotients involving radicals, it is often convenient to move the
radical expression from the denominator to the numerator, or vice versa. For
example, you can move 2 from the denominator to the numerator in the
following quotient by multiplying by 22.
Radical in Denominator
1
2
Rationalize
1 2
2 2
Radical in Numerator
2
2
This process is called rationalizing the denominator. A similar process is used
to rationalize the numerator.
ALGEBRA
Rationalizing Techniques
1. If the denominator is a, multiply by
REVIEW
The success of the second and third
rationalizing techniques stems from
the following.
a .
a
2. If the denominator is a b, multiply by
a b
.
a b
a b a b ab
a b
.
3. If the denominator is a b, multiply by
a b
The same guidelines apply to rationalizing numerators.
EXAMPLE 6
Rationalizing Denominators and Numerators
Rationalize the denominator or numerator.
(a)
3
(b)
12
x 1
(c)
2
1
(d)
5 2
1
x x 1
SOLUTION
(a)
(b)
3233 x1
xx 11 2
3
3
3 3
12
23 23 3
x 1
2
3
2
TRY
x1
2x 1
(c)
(d)
5 2
5 2
5 2
1
1
5 2
5 2 5 2
52
3
1
x x 1
1
x x 1
x x 1 x x 1
x x 1
x x 1
x x 1
IT
6
Rationalize the denominator or
numerator.
5
(a)
8
(b)
x 2
4
1
(c)
6 3
1
(d)
x x 2
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CHAPTER 0
E X E R C I S E S
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0 . 5
In Exercises 1–16, perform the indicated operations and simplify
your answer.
27.
x
x 212
2x 212
28.
x
2x 512
x 512
1.
5
x
x1 x1
2.
2x 1 1 x
x3
x3
29.
x 2
2x
2x 3 32 2x 312
3.
2x
1 3x
2
x2 2
x 2
4.
5x 10 2x 10
2x 1
2x 1
30.
x
3
23 x 232 3 x 212
5.
2
1
x2 4 x 2
6.
x
1
x2 x 2 x 2
In Exercises 31– 44, rationalize the numerator or denominator and
simplify.
7.
5
3
x3 3x
8.
x
2
2x x2
31.
9.
10.
11.
A
B
C
x 1 x 1 2 x 2
33.
A
B
C
x 5 x 5 x 5 2
35.
A
Bx C
2
x6
x 3
12.
1
2
13. 2
x
x 1
15.
14.
Ax B
C
x2 2
x4
37.
2
1x
x 1 x 2 2x 3
39.
1
x
x 2 x 2 x 2 5x 6
41.
x1
2
10
16. 2
x 5x 4 x 2 x 2 x 2 2x 8
In Exercises 17–30, simplify each expression.
x 2 2
18. 2x x 2 2x
2
x
17.
x 1 32 x 112
2t
1 t
19.
2 1t
20. x 2 1
x2
22. 21. 2xx 2 1 1
x 2 1
x3
x 2 1
x 2 1
23.
x 2 212 x 2x 2 212
x2
24.
xx 112 x 112
x2
25.
x
x
x 1
2x 1
2x 2
26.
3x 2 1 23
x 2 113
x2
32.
x
34.
x 4
49x 3
x 2 9
36.
5
14 2
2x
5 3
1
2
10x 2
x 2 x 6
13
6 10
40.
x
2 3
44.
x x 2
4y
y 8
38.
42.
6 5
5
10
15 3
12
10
x x 5
In Exercises 45 and 46, perform the indicated operations and
rationalize as needed.
4 x 2
45.
3x 3
x3 1 x 3 1
2x 3 1
x 1
43.
3
6
x4
2
2
x 4 x2
4 x2
x 2 1
46.
x2
1
xx 2 1
x2 1
47. Installment Loan The monthly payment M for an
installment loan is given by the formula
MP
r12
1
1
r12 1
N
where P is the amount of the loan, r is the annual percentage rate, and N is the number of monthly payments. Enter
the formula into a graphing utility, and use it to find the
monthly payment for a loan of $10,000 at an annual
percentage rate of 14% r 0.14 for 5 years N 60
monthly payments.
48. Inventory A retailer has determined that the cost C of
ordering and storing x units of a product is
C 6x 900,000 .
x
(a) Write the expression for cost as a single fraction.
(b) Determine the cost for ordering and storing x 240
units of this product.