Download A.1 Radicals and Rational Exponents

6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 779
APPENDIX A.1
Radicals and Rational Exponents
779
Appendixes Overview
This section contains a review of some basic algebraic skills. (You should read Section
P.1 before reading this appendix.) Radical and rational expressions are introduced and
radical expressions are simplified algebraically. We add, subtract, and multiply polynomials and factor simple polynomials by a variety of techniques. These factoring techniques are used to add, subtract, multiply, and divide fractional expressions.
A.1 Radicals and Rational
Exponents
What you’ll learn about
• Radicals
• Simplifying Radical Expressions
• Rationalizing the Denominator
• Rational Exponents
... and why
You need to review these
basic algebraic skills if you
don’t remember them.
Radicals
If b 2 = a, then b is a square root of a. For example, both 2 and - 2 are square roots
of 4 because 22 = 1- 222 = 4. Similarly, b is a cube root of a if b 3 = a. For example, 2 is a cube root of 8 because 23 = 8.
DEFINITION Real nth Root of a Real Number
Let n be an integer greater than 1 and a and b real numbers.
1. If b n = a, then b is an nth root of a.
2. If a has an nth root, the principal nth root of a is the nth root having the
same sign as a.
n
The principal nth root of a is denoted by the radical expression 1a. The positive
integer n is the index of the radical and a is the radicand.
Every real number has exactly one real nth root whenever n is odd. For instance, 2 is the
only real cube root of 8. When n is even, positive real numbers have two real nth roots
and negative real numbers have no real nth roots. For example, the real fourth roots of 16
are ⫾2, and -16 has no real fourth roots. The principal fourth root of 16 is 2.
When n = 2, special notation is used for roots. We omit the index and write 1a instead
of 2
2 a. If a is a positive real number and n a positive even integer, its two nth roots are
n
n
denoted by 1a and - 1a.
EXAMPLE 1 Finding Principal nth Roots
(a) 136 = 6 because 62 = 36.
27
3
3 3
27
(b) 3
= because a b =
.
A8
2
2
8
27
3
3 3
27
(c) 3 = - because a - b = - .
A 8
2
2
8
Principal n th Roots
and Calculators
Most calculators have a key for the principal nth
root. Use this feature of your calculator to check
the computations in Example 1.
(d) 2
4 -625 is not a real number because the index 4 is even and the radicand - 625
is negative (there is no real number whose fourth power is negative).
Now try Exercises 7 and 9.
Here are some properties of radicals together with examples that help illustrate their
meaning.
6965_App_pp779-813.qxd
780
1/20/10
3:20 PM
Page 780
APPENDIX A
Caution
Properties of Radicals
Without the restriction that preceded the list,
property 5 would need special attention. For
example,
21-322 Z 11- 322
because 1- 3 on the right is not a real number.
Let u and v be real numbers, variables, or algebraic expressions, and m and n be
positive integers greater than 1. We assume that all of the roots are real numbers
and all of the denominators are not zero.
Property
n
1. 2uv =
2.
Example
n
n
2u # 2v
175 = 125 # 3
= 125 # 13 = 513
n
2
4 96
u
2u
= n
Av
2v
96
= 4
= 2
4 16 = 2
A6
2
46
n
m n
m#n
2#3
32
3 7 = 27 = 2
67
3. 32u = 2 u
4. 12u2n = u
12
4 524 = 5
n
5. 2u m = 12u2m
n
6. 2u n = e
n
2
3 272 = 12
3 2722 = 32 = 9
n
21-622 = ƒ -6 ƒ = 6
ƒ u ƒ n even
u n odd
2
3 1-623 = - 6
Simplifying Radical Expressions
Many simplifying techniques for roots of real numbers have been rendered obsolete because of calculators. For example, when determining the decimal form of 1/12, it was
once very common first to change the fraction so that the radical was in the numerator:
1
1 # 12
12
=
=
2
12
12 12
Using paper and pencil, it was then easier to divide a decimal approximation for 12 by
2 than to divide that decimal into 1. Now either form is quickly computed with a calculator. However, these techniques are still valid for radicals involving algebraic expressions
and for numerical computations when you need exact answers. Example 2 illustrates the
technique of removing factors from radicands.
Properties of Exponents
EXAMPLE 2 Removing Factors from Radicands
Check the Properties of Exponents on page 7 of
Section P.1 to see why
(a) 2
4 80 =
=
=
=
16 = 24 and 9x 4 = 13x 222.
2
4 16 # 5
2
4 24 # 5
2
4 24 # 2
45
22
45
Finding greatest fourth-power factor
16 = 24
Property 1
Property 6
(b) 218x 5 = 29x 4 # 2x
= 213x 2
2 2
# 2x
= 3x 2 22x
Finding greatest square factor
9x4 = 13x222
Properties 1 and 6
(c) 2
4 x 4y 4 = 2
4 1xy24
= ƒ xy ƒ
Finding greatest fourth-power factor
(d) 2
3 -24y 6 = 2
3 1-2y 223 # 3
Finding greatest cube factor
= - 2y 2 2
33
Property 6
Properties 1 and 6
Now try Exercises 29 and 33.
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 781
APPENDIX A.1
Radicals and Rational Exponents
781
Rationalizing the Denominator
The process of rewriting fractions containing radicals so that the denominator is free of
n
radicals is rationalizing the denominator. When the denominator has the form 2u k,
n n-k
multiplying numerator and denominator by 2u
and using property 6 will eliminate
the radical from the denominator because
n
n
n
n
2u k # 2u n - k = 2u k + n - k = 2u n.
Example 3 illustrates the process.
EXAMPLE 3 Rationalizing the Denominator
(a)
(b)
(c)
2
12
12 # 13
16
=
=
=
A3
3
13
13 13
1
1
2
4x
=
# 24 x
3
2
4x 2
4 x3
=
2
4 x3
2
4 x4
2
4 x3
=
ƒxƒ
5 y2
2
5 x 2y 2
2
5 x 2y 2
x2
2
5 x2
2
5 x2 # 2
=
=
=
=
y
B y3
2
5 y3
2
5 y3 2
5 y2
2
5 y5
5
Now try Exercise 37.
Rational Exponents
We know how to handle exponential expressions with integer exponents (see Section P.1).
For example, x 3 # x 4 = x 7, 1x 322 = x 6, x 5/x 2 = x 3, x -2 = 1/x 2, and so forth. But exponents can also be rational numbers. How should we define x 1/2? If we assume that
the same rules that apply for integer exponents also apply for rational exponents we get
a clue. For example, we want
x 1/2 # x 1/2 = x 1.
This equation suggests that x 1/2 = 1x. In general, we have the following definition.
DEFINITION Rational Exponents
Let u be a real number, variable, or algebraic expression, and n an integer
greater than 1. Then
n
u 1/n = 2u.
If m is a positive integer, m/n is in reduced form, and all roots are real numbers,
then
n
u m/n = 1u 1/n2m = 11u2m
and
u m/n = 1u m21/n = 2u m.
n
The numerator of a rational exponent is the power to which the base is raised, and the
denominator is the root to be taken. The fraction m/n needs to be in reduced form because, for instance,
u 2/3 = 12
3 u22
is defined for all real numbers u (every real number has a cube root), but
u 4/6 = 12
6 u24
is defined only for u Ú 0 (only nonnegative real numbers have sixth roots).
6965_App_pp779-813.qxd
782
1/20/10
3:20 PM
Page 782
APPENDIX A
Simplifying Radicals
If you also want the radical form in Example 4d
to be simplified, then continue as follows:
1
1
=
2z 3
#
2z 3
1z
1z
=
1z
z2
EXAMPLE 4 Converting Radicals to Exponentials
and Vice Versa
(a) 21x + y23 = 1x + y23/2
3 x 2y
(c) x 2/3y 1/3 = 1x 2y21/3 = 2
5 x 2 = 3x # x 2/5 = 3x 7/5
(b) 3x2
1
1
(d) z -3/2 = 3/2 =
z
2z 3
Now try Exercises 43 and 47.
An expression involving powers is simplified if each factor appears only once and all
exponents are positive. Example 5 illustrates.
EXAMPLE 5 Simplifying Exponential Expressions
(a) 1x 2y 921/31xy 22 = 1x 2/3y 321xy 22 = x 5/3y 5
(b) a
2x -1/2
6x 1/6
3x 2/3
b
a
b
=
y 1/2
y 2/5
y 9/10
Now try Exercise 61.
Example 6 suggests how to simplify a sum or difference of radicals.
EXAMPLE 6 Combining Radicals
(a) 2180 - 1125 = 2116 # 5 - 125 # 5
= 815 - 515
= 315
Remove factors from radicands.
(b) 24x 2y - 2y 3 = 212x22y - 2y 2y
Find greatest square factors.
= 2 ƒ x ƒ 1y - ƒ y ƒ 1y
= 12 ƒ x ƒ - ƒ y ƒ 21y
Find greatest square factors.
Distributive property
Remove factors from radicands.
Distributive property.
Now try Exercise 71.
Here’s a summary of the procedures we use to simplify expressions involving radicals.
Simplifying Radical Expressions
1. Remove factors from the radicand (see Example 2).
2. Eliminate radicals from denominators and denominators from radicands
(see Example 3).
3. Combine sums and differences of radicals, if possible (see Example 6).
APPENDIX A.1 EXERCISES
In Exercises 1–6, find the indicated real roots.
In Exercises 13–22, use a calculator to evaluate the expression.
1. Square roots of 81
2. Fourth roots of 81
4 256
13. 2
5 3125
14. 2
3. Cube roots of 64
4. Fifth roots of 243
3 15.625
15. 2
16. 112.25
5. Square roots of 16/9
6. Cube roots of -27/8
In Exercises 7–12, evaluate the expression without using a calculator.
7. 1144
3 216
10. 2
8. 1 - 16
64
11. 3 A 27
3 -216
9. 2
64
12.
A 25
3/2
17. 81
18. 165/4
19. 32-2/5
20. 27-4/3
1 -1/3
21. a - b
8
22. a -
125 -1/3
b
64
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 783
APPENDIX A.1
In Exercises 23–26, use the information from the grapher screens below
to evaluate the expression.
1.5^3
61. a
1.3^2
y -3
1x y 2
b
2/3
62.
9 6 -1/3
3.375
4.41^2
-8x 6
Radicals and Rational Exponents
1.69
2.1^4
19.4481
19.4481
63.
1p 2q 421/2
127q 3p 621/3
64. a
1x 6y 22-1/2
783
2x 1/2
y 2/3
ba
3x -2/3
y 1/2
b
In Exercises 65–74, simplify the radical expression.
23. 11.69
24. 119.4481
25. 2
4 19.4481
26. 2
3 3.375
In Exercises 27–36, simplify by removing factors from the radicand.
27. 1288
3 500
28. 2
3 - 250
29. 2
4 192
30. 2
31. 22x 3y 4
3 - 27x 3y 6
32. 2
4 3x 8y 6
33. 2
34.
5 96x 10
35. 2
36. 2108x 4y 9
2
3 8x 6y 4
In Exercises 37–42, rationalize the denominator.
4
1
37.
38.
15
2
3 2
1
2
39.
40.
2
2
5x
2
4y
x2
41. 3
By
a3
42. 5 2
Bb
In Exercises 43–46, convert to exponential form.
3 1a + 2b22
43. 2
3 x 2y
45. 2x 2
5 x 2y 3
44. 2
4 xy 3
46. xy 2
49. x -5/3
48. x 2/3y 1/3
50. 1xy2-3/4
51. 212x
3 3x 2
52. 32
4 1xy
53. 2
3 1ab
54. 2
2
5a
2
3 a2
56. 1a2
2
3a
In Exercises 57–64, simplify the exponential expression.
57.
a 3/5a 1/3
a 3/2
59. 1a 5/3b 3/4213a 1/3b 5/42
58. 1x 2y 421/2
60. a
x 1/2
y
2/3
3x 8y 2
67. 4
B 8x 2
4x 2
2x 2
69. 3 2 # 3
By
B y
4x 6y
68. 5
B 9x 3
5 9ab 6 # 2
5 27a 2b -1
70. 2
71. 3148 - 21108
72. 21175 - 4128
73. 2x - 24xy
74. 218x 2y + 22y 3
3
2
In Exercises 75–82, replace ~ with 6, =, or 7 to make a true statement.
75. 12 + 6 ~ 12 + 16
76. 14 + 19 ~ 14 + 9
77. 13-22-1/2 ~ 3
78. 12-321/3 ~ 2
81. 22/3 ~ 33/4
82. 4-2/3 ~ 3-3/4
4 1- 22 ~ - 2
79. 2
4
3 1- 223 ~ -2
80. 2
83. The time t (in seconds) that it takes for a pendulum to complete
one cycle is approximately t = 1.1 2L, where L is the length
(in feet) of the pendulum. How long is the period of a pendulum of length 10 ft?
84. The time t (in seconds) that it takes for a rock to fall a distance
d (in meters) is approximately t = 0.451d. How long does it
take for the rock to fall a distance of 200 m?
n
In Exercises 51–56, write using a single radical.
55.
66. 216y 8z -2
85. Writing to Learn Explain why 2a and a real nth root
of a need not have the same value.
In Exercises 47–50, convert to radical form.
47. a 3/4b 1/4
65. 29x -6y 4
b
6
6965_App_pp779-813.qxd
3:20 PM
Page 784
APPENDIX A
A.2 Polynomials and Factoring
A polynomial in x is any expression that can be written in the form
• Special Products
• Factoring Trinomials
• Factoring by Grouping
... and why
You need to review these basic
algebraic skills if you don’t
remember them.
where n is a nonnegative integer, and an Z 0. The numbers an - 1, Á , a1, a0 are real
numbers called coefficients. The degree of the polynomial is n and the leading
coefficient is an. Polynomials with one, two, or three terms are monomials,
binomials, or trinomials, respectively. A polynomial written with powers of x in
descending order is in standard form.
To add or subtract polynomials, we add or subtract like terms using the distributive
property. Terms of polynomials that have the same variable each raised to the same
power are like terms.
EXAMPLE 1 Adding and Subtracting Polynomials
(a) 12x 3 - 3x 2 + 4x - 12 + 1x 3 + 2x 2 - 5x + 32
(b) 14x 2 + 3x - 42 - 12x 3 + x 2 - x + 22
SOLUTION
(a) We group like terms and then combine them as follows:
12x 3 + x 32 + 1-3x 2 + 2x 22 + 14x + 1-5x22 + 1-1 + 32
= 3x 3 - x 2 - x + 2
(b) We group like terms and then combine them as follows:
10 - 2x 32 + 14x 2 - x 22 + 13x - 1-x22 + 1 -4 - 22
= - 2x 3 + 3x 2 + 4x - 6
Now try Exercises 9 and 11.
To expand the product of two polynomials we use the distributive property. Here is
what the procedure looks like when we multiply the binomials 3x + 2 and 4x - 5.
13x + 2214x - 52
= 3x14x - 52 + 214x - 52
= 13x214x2 - 13x2152 + 12214x2 - 122152
= 12x 2 - 15x + 8x
- 10
Product of
First terms
Product of Product of
Outer terms Inner terms
Distributive property
Distributive property
兵
• Factoring Polynomials Using
Special Products
anx n + an - 1x n - 1 + Á + a1x + a0,
兵
• Adding, Subtracting,
and Multiplying Polynomials
Adding, Subtracting, and Multiplying
Polynomials
兵
What you’ll learn about
兵
784
1/20/10
Product of
Last terms
In the above FOIL method for products of binomials, the outer (O) and inner (I ) terms
are like terms and can be added to give
13x + 2214x - 52 = 12x 2 - 7x - 10.
Multiplying two polynomials requires multiplying each term of one polynomial by
every term of the other polynomial. A convenient way to compute a product is to
arrange the polynomials in standard form one on top of another so their terms align vertically, as illustrated in Example 2.
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 785
APPENDIX A.2
Polynomials and Factoring
785
EXAMPLE 2 Multiplying Polynomials in Vertical Form
Write 1x 2 - 4x + 321x 2 + 4x + 52 in standard form.
SOLUTION
x 2 - 4x + 3
x 2 + 4x + 5
x 4 - 4x 3 + 3x 2
4x 3 - 16x 2 + 12x
5x 2 - 20x + 15
x 4 + 0x 3 - 8x 2 - 8x + 15
Thus,
= x21x2 - 4x + 32
= 4x1x2 - 4x + 32
= 51x2 - 4x + 32
Add.
1x 2 - 4x + 321x 2 + 4x + 52 = x 4 - 8x 2 - 8x + 15.
Now try Exercise 33.
Special Products
Certain products provide patterns that will be useful when we factor polynomials. Here
is a list of some special products for binomials.
Special Binomial Products
Let u and v be real numbers, variables, or algebraic expressions.
1. Product of a sum and
a difference:
1u + v21u - v2 = u 2 - v 2
2. Square of a sum:
1u + v22 = u 2 + 2uv + v 2
3. Square of a difference:
1u - v22 = u 2 - 2uv + v 2
4. Cube of a sum:
1u + v23 = u 3 + 3u 2v + 3uv 2 + v 3
5. Cube of a difference:
1u - v23 = u 3 - 3u 2v + 3uv 2 - v 3
EXAMPLE 3 Using Special Products
Expand the products.
(a) 13x + 8213x - 82 = 13x22 - 82
= 9x 2 - 64
(b) 15y - 422 = 15y22 - 215y2142 + 42
= 25y 2 - 40y + 16
(c) 12x - 3y23 = 12x23 - 312x2213y2
+ 312x213y22 - 13y23
= 8x 3 - 36x 2y + 54xy 2 - 27y 3
Now try Exercises 23, 25, and 27.
Factoring Polynomials Using Special Products
When we write a polynomial as a product of two or more polynomial factors we are
factoring a polynomial. Unless specified otherwise, we factor polynomials into factors of lesser degree and with integer coefficients in this appendix. A polynomial that
cannot be factored using integer coefficients is a prime polynomial.
6965_App_pp779-813.qxd
786
1/20/10
3:20 PM
Page 786
APPENDIX A
A polynomial is completely factored if it is written as a product of its prime factors.
For example,
2x 2 + 7x - 4 = 12x - 121x + 42
and
x 3 + x 2 + x + 1 = 1x + 121x 2 + 12
are completely factored (it can be shown that x 2 + 1 is prime). However,
x 3 - 9x = x1x 2 - 92
is not completely factored because 1x 2 - 92 is not prime. In fact,
x 2 - 9 = 1x - 321x + 32 and
x 3 - 9x = x1x - 321x + 32
is completely factored.
The first step in factoring a polynomial is to remove common factors from its terms using
the distributive property as illustrated by Example 4.
EXAMPLE 4 Removing Common Factors
(a) 2x 3 + 2x 2 - 6x = 2x1x 2 + x - 32
(b) u v + uv = uv1u + v 2
3
3
2
2
2x is the common factor.
uv is the common factor.
Now try Exercise 43.
Recognizing the expanded form of the five special binomial products will help us factor
them. The special form that is easiest to identify is the difference of two squares. The
two binomial factors have opposite signs:
Two squares
Square roots
u 2 - v 2 = 1u + v21u - v2.
Difference
Opposite signs
EXAMPLE 5 Factoring the Difference of Two Squares
(a) 25x 2 - 36 = 15x22 - 62
= 15x + 6215x - 62
Difference of two squares
Factors are prime.
(b) 4x 2 - 1y + 322 = 12x22 - 1y + 322
= 32x + 1y + 32432x - 1y + 324
= 12x + y + 3212x - y - 32
Difference of two squares
Factors are prime.
Simplify.
Now try Exercise 45.
A perfect square trinomial is the square of a binomial and has one of the two forms
shown here. The first and last terms are squares of u and v, and the middle term is twice
the product of u and v. The operation signs before the middle term and in the binomial
factor are the same.
Perfect square (sum)
Perfect square (difference)
u 2 + 2uv + v 2 = 1u + v22
u 2 - 2uv + v 2 = 1u - v22
Same signs
Same signs
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 787
APPENDIX A.2
Polynomials and Factoring
787
EXAMPLE 6 Factoring Perfect Square Trinomials
(a) 9x 2 + 6x + 1 = 13x22 + 213x2112 + 12
= 13x + 122
u = 3x, v = 1
(b) 4x 2 - 12xy + 9y 2 = 12x22 - 212x213y2 + 13y22
= 12x - 3y22
u = 2x, v = 3y
Now try Exercise 49.
In the sum and difference of two cubes, notice the pattern of the signs.
Same signs
Same signs
u 3 + v 3 = 1u + v21u 2 - uv + v 22
u 3 - v 3 = 1u - v21u 2 + uv + v 22
Opposite signs
Opposite signs
EXAMPLE 7 Factoring the Sum and Difference of Two Cubes
(a) x 3 - 64 = x 3 - 43
= 1x - 421x 2 + 4x + 162
Difference of two cubes
Factors are prime.
(b) 8x 3 + 27 = 12x23 + 33
= 12x + 3214x 2 - 6x + 92
Sum of two cubes
Factors are prime.
Now try Exercise 55.
Factoring Trinomials
Factoring the trinomial ax 2 + bx + c into a product of binomials with integer coefficients requires factoring the integers a and c.
Factors of a
ax 2 + bx + c = 1n x + n21n x + n2
Factors of c
Because the number of integer factors of a and c is finite, we can list all possible binomial factors. Then we begin checking each possibility until we find a pair that works.
(If no pair works, then the trinomial is prime.) Example 8 illustrates.
EXAMPLE 8 Factoring a Trinomial with Leading
Coefficient ⴝ 1
Factor x 2 + 5x - 14.
SOLUTION The only factor pair of the leading coefficient is 1 and 1. The factor
pairs of 14 are 1 and 14, and 2 and 7. Here are the four possible factorizations of the
trinomial:
1x + 121x - 142
1x + 221x - 72
1x - 121x + 142
1x - 221x + 72
If you check the middle term from each factorization you will find that
x 2 + 5x - 14 = 1x - 221x + 72.
Now try Exercise 59.
With practice you will find that it usually is not necessary to list all possible binomial
factors. Often you can test the possibilities mentally.
6965_App_pp779-813.qxd
788
1/20/10
3:20 PM
Page 788
APPENDIX A
EXAMPLE 9 Factoring a Trinomial with Leading
Coefficient ⴝ 1
Factor 35x 2 - x - 12.
SOLUTION The factor pairs of the leading coefficient are 1 and 35, and 5 and 7.
The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. The possible factorizations
must be of the form
1x - *2135x + ?2,
15x - *217x + ?2,
1x + *2135x - ?2,
15x + *217x - ?2,
where * and ? are one of the factor pairs of 12. Because the two binomial factors
have opposite signs, there are 6 possibilities for each of the four forms—a total of
24 possibilities in all. If you try them, mentally and systematically, you should find that
35x 2 - x - 12 = 15x - 3217x + 4).
Now try Exercise 63.
We can extend the technique of Examples 8 and 9 to trinomials in two variables as illustrated in Example 10.
EXAMPLE 10 Factoring Trinomials in x and y
Factor 3x 2 - 7xy + 2y 2.
SOLUTION The only way to get - 7xy as the middle term is with 3x 2 - 7xy +
2y 2 = 13x - ?y21x - ?y2.
The signs in the binomials must be negative because the coefficient of y 2 is positive
and the coefficient of the middle term is negative. Checking the two possibilities,
13x - y21x - 2y2 and 13x - 2y21x - y2, shows that
3x 2 - 7xy + 2y 2 = 13x - y21x - 2y2.
Now try Exercise 67.
Factoring by Grouping
Notice that 1a + b21c + d2 = ac + ad + bc + bd. If a polynomial with four terms
is the product of two binomials, we can group terms to factor. There are only three
ways to group the terms and two of them work. So, if two of the possibilities fail, then
it is not factorable.
EXAMPLE 11 Factoring by Grouping
(a) 3x 3 + x 2 - 6x - 2
= 13x 3 + x 22 - 16x + 22
= x 213x + 12 - 213x + 12
= 13x + 121x 2 - 22
(b) 2ac - 2ad + bc
= 12ac
= 2a1c
= 1c -
- bd
- 2ad2 + 1bc - bd2
- d2 + b1c - d2
d212a + b2
Here is a checklist for factoring polynomials.
Group terms.
Factor each group.
Distributive property
Group terms.
Factor each group.
Distributive property
Now try Exercise 69.
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 789
APPENDIX A.2
Polynomials and Factoring
Factoring Polynomials
1. Look for common factors.
2. Look for special polynomial forms.
3. Use factor pairs.
4. If there are four terms, try grouping.
APPENDIX A.2 EXERCISES
In Exercises 1–4, write the polynomial in standard form and state its
degree.
1. 2x - 1 + 3x
3. 1 - x
2
2
3
2. x - 2x - 2x + 1
7
4. x 2 - x 4 + x - 3
In Exercises 5–8, state whether the expression is a polynomial.
2x - 4
5. x 3 - 2x 2 + x -1
6.
x
7. 1x + x + 12
2
2
8. 1 - 3x + x
9. 1x - 3x + 72 + 13x + 5x - 32
2
10. 1- 3x 2 - 52 - 1x 2 + 7x + 122
12. - 1y + 2y - 32 + 15y + 3y + 42
2
2
13. 2x1x - x + 32
15. - 3u14u - 12
17. 12 - x - 3x 215x2
2
14. y 12y + 3y - 42
2
In Exercises 41–44, factor out the common factor.
41. 5x - 15
42. 5x 3 - 20x
43. yz 3 - 3yz 2 + 2yz
44. 2x1x + 32 - 51x + 32
In Exercises 45–48, factor the difference of two squares.
46. 9y 2 - 16
47. 64 - 25y 2
48. 16 - 1x + 222
In Exercises 49–52, factor the perfect square trinomial.
11. 14x 3 - x 2 + 3x2 - 1x 3 + 12x - 32
2
40. 1x + 121x 2 - x + 12
45. z 2 - 49
4
In Exercises 9–18, simplify the expression. Write your answer in
standard form.
2
39. 1x - 221x 2 + 2x + 42
49. y 2 + 8y + 16
50. 36y 2 + 12y + 1
51. 4z 2 - 4z + 1
52. 9z 2 - 24z + 16
2
16. - 4v12 - 3v 32
18. 11 - x + x 212x2
2
4
In Exercises 19–40, expand the product. Use vertical alignment in
Exercises 33 and 34.
In Exercises 53–58, factor the sum or difference of two cubes.
53. y 3 - 8
54. z 3 + 64
55. 27y 3 - 8
19. 1x - 221x + 52
56. 64z 3 + 27
21. 13x - 521x + 22
58. 27 - y 3
57. 1 - x 3
20. 12x + 3214x + 12
22. 12x - 3212x + 32
In Exercises 59–68, factor the trinomial.
59. x 2 + 9x + 14
23. 13x - y213x + y2
24. 13 - 5x22
60. y 2 - 11y + 30
25. 13x + 4y22
26. 1x - 123
61. z 2 - 5z - 24
29. 12x 3 - 3y212x 3 + 3y2
30. 15x 3 - 122
63. 14u 2 - 33u - 5
27. 12u - v23
31. 1x 2 - 2x + 321x + 42
28. 1u + 3v23
62. 6t 2 + 5t + 1
32. 1x 2 + 3x - 221x - 32
65. 12x 2 + 11x - 15
33. 1x 2 + x - 321x 2 + x + 12
34. 12x 2 - 3x + 121x 2 - x + 22
2
67. 6x + 11xy - 10y
64. 10v 2 + 23v + 12
66. 2x 2 - 3xy + y 2
2
68. 15x 2 + 29xy - 14y 2
In Exercises 69–74, factor by grouping.
35. 1x - 1221x + 122
69. x 3 - 4x 2 + 5x - 20
70. 2x 3 - 3x 2 + 2x - 3
36. 1x 1/2 - y 1/221x 1/2 + y 1/22
71. x 6 - 3x 4 + x 2 - 3
72. x 6 + 2x 4 + x 2 + 2
37. 1 1u + 1v21 1u - 1v2
73. 2ac + 6ad - bc - 3bd
38. 1x 2 - 1321x 2 + 132
74. 3uw + 12uz - 2vw - 8vz
789
6965_App_pp779-813.qxd
790
1/20/10
3:20 PM
Page 790
APPENDIX A
In Exercises 75–90, factor completely.
3
2
3
2
76. 4y - 20y + 25y
75. x + x
3
91. Writing to Learn Show that the grouping
3
2
77. 18y + 48y + 32y
78. 2x - 16x + 14x
79. 16y - y 3
80. 3x 4 + 24x
81. 5y + 3y 2 - 2y 3
82. z - 8z 4
83. 215x + 122 - 18
84. 512x - 322 - 20
85. 12x 2 + 22x - 20
86. 3x 2 + 13xy - 10y 2
87. 2ac - 2bd + 4ad - bc 88. 6ac - 2bd + 4bc - 3ad
89. x 3 - 3x 2 - 4x + 12
90. x 4 - 4x 3 - x 2 + 4x
12ac + bc2 - 12ad + bd2
leads to the same factorization as in Example 11b. Explain why
the third possibility,
12ac - bd2 + 1 -2ad + bc2
does not lead to a factorization.
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 791
APPENDIX A.3
Fractional Expressions
791
A.3 Fractional Expressions
What you’ll learn about
• Domain of an Algebraic
Expression
• Reducing Rational Expressions
• Operations with Rational
Expressions
• Compound Rational
Expressions
... and why
You need to review these basic
algebraic skills if you don’t
remember them.
Domain of an Algebraic Expression
A quotient of two algebraic expressions, besides being another algebraic expression,
is a fractional expression, or simply a fraction. If the quotient can be written as the
ratio of two polynomials, the fractional expression is a rational expression. Here are
examples of each.
x 2 - 5x + 2
2x 2 + 1
2x 3 - x 2 + 1
5x 2 - x - 3
The one on the left is a fractional expression but not a rational expression. The other is
both a fractional expression and a rational expression.
Unlike polynomials, which are defined for all real numbers, some algebraic expressions are not defined for some real numbers. The set of real numbers for which an algebraic expression is defined is the domain of the algebraic expression.
EXAMPLE 1 Finding Domains of Algebraic Expressions
(a) 3x 2 - x + 5
(b) 1x - 1
(c)
x
x - 2
SOLUTION
(a) The domain of 3x 2 - x + 5, like that of any polynomial, is the set of all real
numbers.
(b) Because only nonnegative numbers have square roots, x - 1 Ú 0, or x Ú 1. In
interval notation, the domain is 31, q 2.
(c) Because division by zero is undefined, x - 2 Z 0, or x Z 2. The domain is the
set of all real numbers except 2.
Now try Exercises 11 and 13.
Reducing Rational Expressions
Let u, v, and z be real numbers, variables, or algebraic expressions. We can write rational
expressions in simpler form using
u
uz
=
vz
v
provided z Z 0. This requires that we first factor the numerator and denominator into
prime factors. When all factors common to numerator and denominator have been
removed, the rational expression (or rational number) is in reduced form.
EXAMPLE 2 Reducing Rational Expressions
Write 1x 2 - 3x2/1x 2 - 92 in reduced form.
SOLUTION
x1x - 32
x 2 - 3x
=
2
1x + 321x - 32
x - 9
x
=
, x Z 3
x + 3
Factor completely.
Remove common factors.
We include x Z 3 as part of the reduced form because 3 is not in the domain of the
original rational expression and thus should not be in the domain of the final rational
expression.
Now try Exercise 35.
6965_App_pp779-813.qxd
792
1/20/10
3:20 PM
Page 792
APPENDIX A
Two rational expressions are equivalent if they have the same domain and have the
same value for all numbers in the domain. The reduced form of a rational expression
must have the same domain as the original rational expression. This is why we attached
the restriction x Z 3 to the reduced form in Example 2.
Operations with Rational Expressions
Two fractions are equal, u/v = z /w, if and only if uw = vz. Here is how we operate
with fractions.
Operations with Fractions
Invert and Multiply
The division step shown in 4 is often referred to
as invert the divisor (the fraction following the
division symbol) and multiply the result times
the numerator (the first fraction).
Let u, v, w, and z be real numbers, variables, or algebraic expressions. All of the
denominators are assumed to be different from zero.
Operation
Example
u
w
u + w
5
2 + 5
7
2
1.
+
=
+ =
=
v
v
v
3
3
3
3
u
2
4
2#5 + 3#4
22
w
uz + vw
+ =
=
+
=
2.
#
v
z
vz
3
5
3 5
15
u#w
uw
2#4
2#4
8
=
= # =
3.
v z
vz
3 5
3 5
15
2
u
w
u# z
uz
4
2#5
10
5
,
=
=
, =
=
=
4.
v
z
v w
vw
3
5
3 4
12
6
5. For subtraction, replace “ + ” by “ - ” in 1 and 2.
EXAMPLE 3 Multiplying and Dividing Rational Expressions
(a)
(b)
2x 2 + 11x - 21 #
x3 - 8
x 3 + 2x 2 + 4x x 2 + 5x - 14
12x - 321x + 72 1x - 221x 2 + 2x + 42
#
=
1x - 221x + 72
x1x 2 + 2x + 42
2x - 3
=
, x Z 2, x Z - 7
x
x3 + 1
x2 - x + 1
,
x2 - x - 2
x 2 - 4x + 4
1x 3 + 121x 2 - 4x + 42
= 2
1x - x - 221x 2 - x + 12
=
1x + 121x 2 - x + 121x - 222
x Z - 1,
Remove common factors.
Invert and multiply.
1
1x + 121x - 221x 2 - x + 12
= x - 2,
Factor completely.
x Z 2
Factor completely.
Remove common factors.
Now try Exercises 49 and 55.
Note on Example
2
The numerator, x + 4x - 6, of the final
expression in Example 4 is a prime polynomial.
Thus, there are no common factors.
EXAMPLE 4 Adding Rational Expressions
x1x - 52 + 313x - 22
x
3
+
=
3x - 2
x - 5
13x - 221x - 52
=
=
x 2 - 5x + 9x - 6
13x - 221x - 52
x 2 + 4x - 6
13x - 221x - 52
Definition of addition
Distributive property
Combine like terms.
Now try Exercise 59.
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 793
APPENDIX A.3
Fractional Expressions
793
If the denominators of fractions have common factors, then it is often more efficient
to find the LCD before adding or subtracting the fractions. The LCD (least common
denominator) is the product of all the prime factors in the denominators, where each
factor is raised to the greatest power found in any one denominator for that factor.
EXAMPLE 5 Using the LCD
Write the following expression as a fraction in reduced form.
2
1
3
+ - 2
2
x
x - 2x
x - 4
SOLUTION The factored denominators are x1x - 22, x, and 1x - 221x + 22,
respectively. The LCD is x1x - 221x + 2).
2
1
3
+ - 2
x
x 2 - 2x
x - 4
2
1
3
=
+ x
x1x - 22
1x - 221x + 22
21x + 22
=
=
x1x - 221x + 22
+
1x - 221x + 22
x1x - 221x + 22
21x + 22 + 1x - 221x + 22 - 3x
Factor.
-
3x
x1x - 221x + 22
Equivalent
fractions
Combine
numerators.
x1x - 221x + 22
=
2x + 4 + x 2 - 4 - 3x
x1x - 221x + 22
Expand terms.
=
x2 - x
x1x - 221x + 22
Simplify.
x1x - 12
Factor.
=
x1x - 221x + 22
x - 1
=
, x Z 0
1x - 221x + 22
Reduce.
Now try Exercise 61.
Compound Rational Expressions
Sometimes a complicated algebraic expression needs to be changed to a more familiar
form before we can work on it. A compound fraction (sometimes called a complex fraction), in which the numerators and denominators may themselves contain fractions, is such
an example. One way to simplify a compound fraction is to write both the numerator and
denominator as single fractions and then invert and multiply. If the fraction then takes the
form of a rational expression, we write the expression in reduced or simplest form.
EXAMPLE 6 Simplifying a Compound Fraction
31x + 22 - 7
7
x + 2
x + 2
=
1
1x - 32 - 1
1 x - 3
x - 3
3x - 1
x + 2
=
x - 4
x - 3
13x - 121x - 32
=
,
1x + 221x - 42
3 -
Combine fractions.
Simplify.
x Z 3
Invert and multiply.
Now try Exercise 63.
6965_App_pp779-813.qxd
794
1/20/10
3:20 PM
Page 794
APPENDIX A
A second way to simplify a compound fraction is to multiply the numerator and denominator by the LCD of all fractions in the numerator and denominator as illustrated in
Example 7.
EXAMPLE 7 Simplifying Another Compound Fraction
Use the LCD to simplify the compound fraction
1
1
- 2
2
a
b
.
1
1
a
b
SOLUTION The LCD of the four fractions in the numerator and denominator is
a 2b 2.
1
1
1
1
a 2 - 2 ba 2b 2
- 2
2
a
b
a
b
=
1
1
1
1
a - ba 2b 2
a
b
a
b
=
=
b 2 - a2
ab 2 - a 2b
1b + a21b - a2
ab1b - a2
b + a
=
, a Z b
ab
Multiply numerator and
denominator by LCD.
Simplify.
Factor.
Reduce.
Now try Exercise 69.
APPENDIX A.3 EXERCISES
In Exercises 1–8, rewrite as a single fraction.
5
10
+
9
9
20 # 9
3.
21 22
2
4
,
5.
3
5
1
4
5
+
7.
14
15
21
1.
17
9
32
32
33 # 20
4.
25 77
9
15
,
6.
4
10
6
4
1
+
8.
6
35
15
2.
In Exercises 9–18, find the domain of the algebraic expression.
9. 5x 2 - 3x - 7
11. 1x - 4
13.
14.
2x + 1
x 2 + 3x
x2 - 2
x2 - 4
x
, x Z 2
15.
x - 1
3x - 1
, x Z 0
16.
x - 2
2
17. x + x
-1
18. x1x + 12
-2
10. 2x - 5
2
12.
1x + 3
In Exercises 19–26, find the missing numerator or denominator so that
the two rational expressions are equal.
15y
2
?
5
19.
20.
=
=
3
3x
2y
?
12x
21.
x - 4
x 2 - 4x
=
x
?
23.
x + 3
?
= 2
x - 2
x + 2x - 8
24.
x 2 - x - 12
x - 4
=
x + 5
?
22.
x
?
= 2
x + 2
x - 4
x 2 - 3x
x - 3
= 2
?
x + 2x
x2 + x - 6
?
=
26. 2
x - 3
x - 9
25.
In Exercises 27–32, consider the original fraction and its reduced form
from the specified example. Explain why the given restriction is needed
on the reduced form.
27. Example 3a, x Z 2, x Z - 7
28. Example 3b, x Z - 1, x Z 2
29. Example 4, none
30. Example 5, x Z 0
31. Example 6, x Z 3
32. Example 7, a Z b
6965_App_pp779-813.qxd
1/20/10
3:20 PM
Page 795
APPENDIX A.3
In Exercises 33–44, write the expression in reduced form.
3
33.
35.
37.
18x
15x
34.
75y
57.
2y 2 + 6y
36.
4y + 12
x3
2
x - 2x
z 2 - 3z
38.
9 - z2
40.
41.
x 2 - x - 12
y - y - 30
y 2 - 3y - 18
y 3 + 4y 2 - 21y
y 2 - 49
8z 3 - 1
2z 2 + 5z - 3
3
43.
42.
2z 3 + 6z 2 + 18z
z 3 - 27
y 2 + 3y
2
x + 2x - 3x - 6
x 3 + 2x 2
44.
y 3 + 3y 2 - 5y - 15
In Exercises 45–62, simplify.
45.
46.
47.
49.
x
x
x
x
3
+
7
+
-
#x
2
- 1
1
9
3 # 14
2x + 6
3# 1 - x
1 x2 - 9
x3 - 1 #
4x
2x 2
x2 + x + 1
2
51.
2y + 9y - 5
#
y 2 - 25
1
1
,
53.
2x
4
55.
48.
2xy
x 2 - 3x
,
14y
3y 2
y - 5
2y 2 - y
50.
18x 2 - 3x # 12y 2
3xy
6x - 1
y 3 + 2y 2 + 4y y 2 - 4
#
y 3 + 2y 2
1x - 322
8xy
x - 3
58.
x 2 - y2
2xy
y2 - x 2
4x 2y
2x + 1
3
x + 5
x + 5
3
x + 1
+
60.
x - 2
x - 2
1
3
6
- - 2
61. 2
x
x + 3x
x - 9
2
5
4
+ 2
62. 2
x - 2
x + x - 6
x - 4
In Exercises 63–70, simplify the compound fraction.
y
x
1
1
- 2
+
2
y
x
x
y
63.
64.
1
1
1
1
- 2
- 2
2
2
y
x
x
y
13
13x - 3
2 2x +
x - 4
x + 5
65.
66.
x + 3
3
2x +
2 +
x - 4
x - 3
1
1
x
x + h
- 2
1x + h22
x
x + h + 2
x + 2
67.
68.
h
h
b
1
a
1
+
a
a
b
b
69.
70.
1
b
1
a
a
a
b
b
59.
x 2 + 6x + 9
2
39.
2x 2y
2
9y 4
Fractional Expressions
y3 - 8
2
52.
y + 8y + 16 3y 2 + 2y
#
y + 4
3y 2 - y - 2
54.
8y
4x
,
y
x
56.
14x - 14y
7x - 7y
,
4y
3y
In Exercises 71–74, write with positive exponents and simplify.
1x + y2-1
1
1
71. a + b1x + y2-1
72.
x
y
1x - y2-1
73. x -1 + y -1
74. 1x -1 + y -12-1
795