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Barnett/Ziegler/Byleen
Precalculus: A Graphing Approach
Appendix A
Basic Algebra Review
Copyright © 2000 by the McGraw-Hill Companies, Inc.
The Set of Real Numbers
N
Natural Numbers
1, 2, 3, . . .
Z
Integers
. . . , –2, –1, 0, 1, 2, . . .
Rational Numbers
__
–3 2
–4, 0, 8, 5 , 3 , 3.14, –5.2727
Q
I
R
Irrational Numbers
Real Numbers
2, 
3
–7, 0,
3
7 , 1.414213 . . .
–2
–
5 , 3 , 3.14, 0.333 ,
Copyright © 2000 by the McGraw-Hill Companies, Inc.

A-1-113
Subsets of the Set of Real Numbers
Natural
numbers (N)
Zero
Integers (Z)
Rational
numbers (Q)
Negatives
of natural
numbers
Noninteger
ratios
of integers
N 
Z 
Q 
Real
numbers (R)
Irrational
numbers (I)
R
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-1-114
Basic Real Number Properties
Let R be the set of real numbers and let x, y, and z be arbitrary
elements of R.
Addition Properties
Closure:
x + y is a unique element in R.
Associative:
Commutative:
Identity:
Inverse:
(x + y ) + z = x + (y + z )
x +y =y +x
0 +x = x + 0 =x
x + (–x ) = (–x ) + x = 0
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-1-115(a)
Basic Real Number Properties
Multiplication Properties
Closure:
xy is a unique element in R .
Associative:
(xy)z = x(yz)
Commutative:
xy = yx
Identity:
(1) x = x (1) = x
Inverse:
X 
1
1
= 
 x
 x
x =1x 0
Combined Property
Distributive:
x (y + z ) = xy + xz
( x + y ) z = xz + yz
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-1-115(b)
Foil Method
F
(2x – 1)(3x + 2)
O
I
L
First
Outer
Inner
Last
Product Product Product Product
= 



–
–
6x2
+
4x
3x
2
Special Products
2
1. (a – b)(a + b) = a
–
2
b
2. (a + b)2 = a2 + 2ab + b2
3. (a – b)2 = a2 – 2ab + b2
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-2-116
Special Factoring Formulas
1.
u 2 + 2 uv
+ v 2 = ( u + v )2
Perfect Square
2.
u 2 – 2uv + v 2 = (u – v) 2
Perfect Square
3.
u 2 – v 2 = (u – v)(u + v)
Difference of Squares
4.
u 3 – v 3 = (u – v)(u 2 + uv + v 2)
Difference of Cubes
5.
u 3 + v 3 = (u + v)(u2 – uv + v 2)
Sum of Cubes
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-3-117
The Least Common Denominator
(LCD)
The LCD of two or more rational expressions is found as follows:
1. Factor each denominator completely.
2. Identify each different prime factor from all the denominators.
3. Form a product using each different factor to the highest power
that occurs in any one denominator. This product is the LCD.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-4-118
Definition of an
1. For n a positive integer:
an = a · a · … · a
n factors of a
Exponent Properties
1. a m a n = a m + n
2. ( a n) m = a mn
3. (ab)m = a m bm
2. For n = 0 ,
a0 = 1 a  0
00 is not defined
am
am


4.
= m
b
b
b0
3. For n a negative integer,
1
an = –n
a
a 0
am
1
5. n = a m–n = n–m
a
a
Copyright © 2000 by the McGraw-Hill Companies, Inc.
a0
A-5-119
Definition of b1/n
For n a natural number and b a real number,
b1/n is the principal nth root of b
defined as follows:
1. If n is even and b is positive, then b1/n represents the positive nth root of b.
2. If n is even and b is negative, then b1/n does not represent a real number.
3. If n is odd, then b1/n represents the real nth root of b (there is only one).
4. 01/n = 0
Rational Exponents


b
=  ( b m )1/n


Copyright © 2000 by the McGraw-Hill Companies, Inc.
For m and n natural numbers and b any real number
(except b cannot be negative when n is even):
( b1/ n ) m
m/n
A-6-120
n
b , nth-Root Radical
For n a natural number greater than 1 and b a real number, we define
be the principal nth root of b; that is,
n
If n = 2, we write b in place of
2
n
b to
b = b1/n
b.
Rational Exponent/
Radical Conversions
For m and n positive integers (n > 1), and b not negative when n is even,
1.
2.
n
n
(bm)1/n = n bm
bm/n = 
n m
1/n
m
(b
)
=
(
b)

xn = x
xy =
n
x
n x nx
3.
y =n
y
n
y
Properties of Radicals
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-7-121
Simplified (Radical) Form
1. No radicand (the expression within the radical sign) contains a
factor to a power greater than or equal to the index of the radical.
(For example, x5 violates this condition.)
2. No power of the radicand and the index of the radical have a
common factor other than 1.
(For example,
6
x4 violates this condition.)
3. No radical appears in a denominator.
y
(For example,
violates this condition.)
x
4. No fraction appears within a radical.
3
(For example,
5 violates this condition.)
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-7-122
Interval Notation
Interval
Notation
Inequality
Notation
[a, b]
a x b
[a, b)
a x<b
(a, b]
a<x b
(a, b)
a<x<b
Line Graph
Type
[
]
x
Closed
[
)
x
Half-open
(
]
x
Half-open
(
)
x
Open
a
a
a
a
b
b
b
b
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-8-123(a)
Interval Notation
Interval
Notation
[b ,  )
( b,  )
( –, a]
( –, a)
Inequality
Notation
Line Graph
Type
x b
[
b
x
x
x> b
(
b
x a
x< a
Closed
Open
]
a
x
)
a
x
Copyright © 2000 by the McGraw-Hill Companies, Inc.
Closed
Open
A-8-123(b)
Inequality Properties
For a, b, and c any real numbers:
1. If a < b and b < c, then a < c.
Transitive Property
2. If a < b, then a + c < b + c.
Addition Property
3. If a < b, then a – c < b – c.
Subtraction Property


5. If a < b and c is negative, then ca > cb. 
Multiplication Property
(Note difference between
4 and 5.)



Division Property
(Note difference between
6 and 7.)
4. If a < b and c is positive, then ca < cb.
a b
6. If a < b and c is positive, then c < .
c
a b
7. If a < b and c is negative, then c > c .
Copyright © 2000 by the McGraw-Hill Companies, Inc.
A-8-124
Partial Fraction Decomposition
Any proper fraction P(x)/D(x) reduced to lowest terms can be decomposed in the sum of partial
fractions as follows:
1. If D(x) has a nonrepeating linear factor of the form ax + b, then the partial fraction decomposition
of P(x)/D(x) contains a term of the form
A
ax + b A a constant
2. If D(x) has a k-repeating linear factor of the form (ax + b)k, then the partial fraction decomposition
of P(x)/D(x) contains k terms of the form
A1
A2
Ak
A1 , A2 , …, Ak constants
ax + b + (ax + b)2 + … + (ax + b)k
3. If D(x) has a nonrepeating quadratic factor of the form ax2 + bx + c, which is prime relative to the
real numbers, then the partial fraction decomposition of P(x)/D(x) contains a term of the form
Ax + B
ax 2 + bx + c
A, B constants
4. If D(x) has a k-repeating quadratic factor of the form (ax2 + bx + c)k, where ax2 + bx + c is prime
relative to the real numbers, then the partial fraction decomposition of P(x)/D(x) contains k terms of
the form
A1 x + B1
A2 x + B2
Ak x + Bk
+
+…+
(ax2 + bx + c)k
ax2 + bx + c (ax2 +bx + c)2
A1 , …, Ak , B1 , …, Bk constants
Copyright © 2000 by the McGraw-Hill Companies, Inc.
B-1-125
Significant Digits
If a number x is written in scientific notation as
x = a  10n
1  a < 10 , n an integer
then the number of significant digits in x is the number of digits in a.
The number of significant digits in a number with no decimal point if
found by counting the digits from left to right, starting with the first
digit and ending with the last nonzero digit.
The number of significant digits in a number containing a decimal
point is found by counting the digits from left to right, starting with
the first nonzero digit and ending with the last digit.
Rounding Calculated Values
The result of a calculation is rounded to the same number of
significant digits as the number used in the calculation that has the
least number of significant digits.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
C-1-126