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CHAPTER R
SECTION R.1
DETAILED SUMMARY
Sets and the Real Number Line
KEY CONCEPTS
EXAMPLES
Important subsets of the set of real numbers:
Natural Numbers:
⺞ {1, 2, 3, …}
Whole Numbers:
⺧ {0, 1, 2, 3, …}
Integers:
⺪ {… , 3, 2, 1, 0, 1, 2, 3, …}
Example 1
2
3 僆⺡
Rational Numbers:
⺪ ( ⺡
⺡
E pq
0 p, q苸⺪ and q ⬆ 0F
2 僆 ⺞
2
3
is an element of the set of rational numbers.
2 is not an element of the set of natural
numbers.
The set of integers is a proper subset of the set
of rational numbers.
Irrational Numbers: ⺘ is the set of real numbers that cannot
be expressed as a ratio of integers.
Example 2
Set-builder
notation
Graph
{x 0 x 3}
{x 0 2 x 5}
Absolute value:
For a real number x, the absolute value of x is
0x0 e
x
x
if x 0
if x 0
Interval
notation
(
Set-builder and interval notation:
An interval on the real number line can be represented in
set-builder notation or in interval notation.
(, 3)
3
(
(2, 5]
22
Example 3
0 2 17 0 A2 17B
5
because 2 17 0.
17 2
Distance between points on a number line:
The distance between two points a and b on a number line is
given by 0 a b0 or 0 b a0 .
Example 4
The distance between 6 and 4 on the number line is
0 6 (4) 0 10 units or 0 4 6 0 10 units.
Order of operations:
1. Simplify expressions within grouping symbols. If nested
grouping symbols are present, start with the innermost
symbols.
Example 5
Simplify.
2. Evaluate expressions involving exponents.
3. Perform multiplication or division from left to right.
4. Perform addition or subtraction from left to right.
12 2(5 8)2 252 42
12 2(3) 2 125 16
12 2(9) 19
12 18 3
6 3
3
SECTION R.2
Models, Algebraic Expressions, and Properties of Real Numbers
KEY CONCEPTS
EXAMPLES
A formula relating two or more variables is one type of
mathematical model.
Example 1
In the province of Saskatchewan, Canada, an individual
caught speeding between 31–50 km/hr (inclusive) over the
limit would pay a $70 surcharge plus $2 for every km/hr over
the posted limit.
The cost C (in $) for a ticket in this situation is given by:
C 70 2x
where x is the number of km/hr above the
posted speed limit.
Example 2
Use the model from Example 1 to determine the cost of a
speeding ticket for an individual traveling 40 km/hr over the
posted speed limit.
C 70 2x
C 70 2(40)
70 80
150
Substitute 40 km/hr for x.
The driver would pay $150.
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Chapter R
Review of Prerequisites
Properties of real numbers:
• Commutative property of addition
• Commutative property of multiplication
Example 3
abba
aⴢbbⴢa
• Associative property of addition
• Associative property of multiplication
(a b) c a (b c)
(a ⴢ b) ⴢ c a ⴢ (b ⴢ c)
• Identity property of addition
• Identity property of multiplication
a 0 a and 0 a a
a ⴢ 1 a and 1 ⴢ a a
• Inverse property of addition
• Inverse property of multiplication
a (a) 0 and (a) a 0
a ⴢ 1a 1 and 1a ⴢ a 1 where a ⬆ 0
• Distributive property of multiplication over addition
a ⴢ (b c) a ⴢ b a ⴢ c
Simplifying expressions:
Simplify an expression by applying the distributive property
to clear parentheses. Then combine like terms.
SECTION R.3
Example 4
Simplify.
4(8 y) 2(3y 9) 1
32 4y 6y 18 1
10y 15
Integer Exponents and Scientific Notation
KEY CONCEPTS
EXAMPLES
Definition of b0 and bⴚn:
If b is a nonzero real number and n is an integer, then
Example 1
(4p)0 1
1
w6 6
w
b0 1
and
bn 1
bn
Properties of exponents:
• Product rule for exponents
Example 2
bm ⴢ bn bmn
bm
bmn
bn
x7 ⴢ x2 x7(2) x5
z9
z95 z4
z5
• Power rule for exponents
(bm)n bmⴢn
(y2 ) 4 y2ⴢ4 y8 • Power of a product
(ab)m ambm
(2x)4 24 ⴢ x4 16x4
• Power of a quotient
a m am
a b m
b
b
5 3
53
125
a 2b 2 3 6
x
1x 2
x
• Quotient rule for exponents
Scientific notation:
A number expressed in the form a 10n, where 1 0a0 10
and n is an integer is said to be in scientific notation.
1
y8
Example 3
54,000 5.4 104
0.000 000 000 568 5.68 1010
8.75 8.75 100
Example 4
(2.5 108)(6.0 103) (2.5)(6.0) (108)(103)
15 1011
(1.5 101 ) 1011
1.5 1012
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Detailed Summary
SECTION R.4
Rational Exponents and Radicals
KEY CONCEPTS
EXAMPLES
Definition of an nth-root:
b is an nth-root of a if bn a.
Example 1
7 is a square root of 49 because (7)2 49.
7 is a square root of 49 because (7)2 49.
149 7 because 7 0 and (7)2 49.
n
1a represents the principal nth-root of a.
Definition of a1兾n and am兾n:
n
If n 1 is an integer and 1a is a real number, then
n
• a1兾n 1a
n
• am兾n A 1aB m
n
am兾n 2am
and
Properties of rational exponents:
The properties of integer exponents learned in Section R.3
can be extended to rational exponents, provided that all roots
represent real numbers.
Example 2
3
(64)1兾3 1
64 4
4
1兾4
(16) is undefined because 1
16 is not a real number.
3
2兾3
2
2
125 (1125) (5) 25
Example 3
a
x1兾5x2兾5
x4兾5
b
10
a
x3兾5
x
b
4兾5
10
(x3兾54兾5 ) 10
(x1兾5 ) 10 x(1兾5)(10) x2 Simplifying radicals:
Let n 1 be an integer and a be a real number.
• If n is even then 2a 0 a 0 .
n
• If n is odd then 2an a.
n
n
n
n
n
n
m
1a
a
n .
n
1b B b
n
Example 4
6 6
2
y 0y0
and
5 5
2
y y
3 13
3 12
3 12
3
3
2
x 2
x ⴢx 2
x ⴢ1
x x4 1
x
Product property of radicals: 1a ⴢ 1b 2ab.
Quotient property of radicals:
mⴢn
Property of nested radicals: 2 1a 1a.
3
2
2x4
3
116x
2x4
x3
x
3
B 16x B 8
2
3
5 3
15
2
1x 1
x
Operations on radicals:
To multiply radicals, apply the product property of radicals.
Example 5
3
3
3
3
2
4x2 ⴢ 1
6x 2
24x3 2x 1
3
To add or subtract like radicals apply the distributive
property.
423x3 x 112x 5x 13x
4x 13x 2x 13x 5x 13x
(4 2 5)x 13x
x 13x
SECTION R.5
1
x2
Simplify each radical,
then combine like
radicals.
Polynomials and Multiplication of Radicals
KEY CONCEPTS
EXAMPLES
Definition of a polynomial:
A polynomial in the variable x is an expression of the form:
anxn an1xn1 an2xn2 p a1x a0
Example 1
9x5 5x7 3x 4 can be written in descending order as
5x7 9x5 3x (4).
The coefficients an, an1, an2, … , a0 are real numbers where
an ⬆ 0, and the exponents are whole numbers. The term anxn
is the leading term, and an is the leading coefficient.
The leading term is 5x7 and the leading coefficient is 5.
The degree of the polynomial is 7.
The degree of a term is the sum of the exponents on the
variable factors. The degree of the polynomial is the same as
the degree of the leading term.
The degree of the term 6a7b4c is 12 because the sum of the
exponents 7 4 1 is 12.
Addition and subtraction of polynomials:
To add or subtract polynomials, combine like terms.
Example 2
(7x2 3x 6) (2x2 8x)
7x2 3x 6 (2x2) 8x
5x2 5x 6
3
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Chapter R
Review of Prerequisites
Multiplication of polynomials:
To multiply polynomials and radical expressions, use the
distributive property to multiply each term in the first
expression by each term in the second.
Example 3
Special case products:
(a b)(a b) a2 b2
Example 4
(2x 7)(2x 7) (2x)2 (7)2 4x2 49
(a b)2 a2 2ab b2
(3x 2)2 (3x)2 2(3x)(2) (2)2
9x2 12x 4
A51x 3B 2 A51xB 2 2A51xB(3) (3)2
25x 301x 9
(a b)2 a2 2ab b2
SECTION R.6
(3y 4)(2y 5) 6y2 15y 8y 20
6y2 7y 20
21xA31x 51z 4B
21xA31xB 21xA51zB 21x(4)
6x 101xz 81x
Factoring
KEY CONCEPTS
EXAMPLES
General factoring strategy:
1. Factor out the GCF.
2. Identify the number of terms.
4 terms: Factor by grouping either 2 terms with 2 terms
or 3 terms with 1 term.
3 terms: If the trinomial is a perfect square trinomial,
factor as the square of a binomial.
Example 1
30ax 150a 20cx 100c
10[3ax 15a 2cx 10c]
10[3a(x 5) 2c(x 5)]
10(x 5)(3a 2c)
a 2ab b (a b)
a2 2ab b2 (a b)2
2
2
GCF
Grouping
Perfect square trinomial: 9x2 6x 1 (3x 1)2
2
Otherwise, factor by trial-and-error.
2 terms: Determine whether the binomial fits one of the
following patterns:
a2 b2 (a b)(a b)
a3 b3 (a b)(a2 ab b2)
a3 b3 (a b)(a2 ab b2)
Factoring using substitution:
Sometimes we make an appropriate substitution to convert
a cumbersome expression to one that is more easily
recognizable and factorable.
2x2 11x 5 ⱨ (2x 5)(x 1)
2x2 11x 5 ⱨ (2x 1)(x 5) ✓
Difference of squares: x2 y4 (x y2)(x y2)
Sum of cubes: 8z3 27 (2x 3)(4x2 6x 9)
Difference of cubes:
125y6 1 (5y2 1)(25y4 5y2 1)
Example 2
y2 10y 25 z2
(y 5)2 z2
u2 z2
(u z)(u z)
(y 5 z)(y 5 z)
Factoring out negative and rational exponents:
To factor out a common variable factor raised to a negative or
rational exponent, factor out the variable raised to the
smallest exponent.
The exponents on the variables within parentheses are found
by subtraction.
No
Yes
Group 3 terms with 1 term.
The first 3 terms form a perfect
square trinomial.
Substitute u y 5.
Factor as a difference of squares.
Back substitute.
8 (8) 7 (8) 6 (8)
Example 3
7z8 6z7 5z6 z8(7z0 6z1 5z2)
7 6z 5z2
z8(7 6z 5z2) or
z8
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Detailed Summary
SECTION R.7
Rational Expressions and More Operations on Radicals
KEY CONCEPTS
EXAMPLES
Definition of a rational expression:
A rational expression is a ratio of two polynomials. Values
of the variable that make the denominator equal to zero are
called restricted values of the variable.
Example 1
x5
is a rational expression with the restriction
x7
that x ⬆ 7.
Simplify rational expressions:
To simplify a rational expression, use the property of
equivalent algebraic fractions.
Example 2
ac
a
for b ⬆ 0, c ⬆ 0
bc
b
1
(2x 1)(x 5)
2x2 9x 5
2
6x(x 5)
6x 30x
2x 1
for x ⬆ 0, x ⬆ 5
6x
Multiply and divide rational expressions:
a c
ac
ⴢ for b ⬆ 0, d ⬆ 0
b d
bd
a
c
a d
ⴢ for b ⬆ 0, c ⬆ 0, d ⬆ 0
b
d
b c
Example 3
12 3x
x2 3x 4
2
2x 1
2x 15x 7
Add and subtract rational expressions:
Write each fraction as an equivalent fraction with a common
denominator. Then apply the following properties.
Example 4
x(x)
2(x 1)
x
2
x
x1
(x 1)(x)
x(x 1)
x2 2(x 1)
x2 2x 2
x(x 1)
x(x 1)
a
c
ac
a
c
ac
and for b ⬆ 0
b
b
b
b
b
b
(1)
1
3(4 x) (2x 1)(x 7)
3(x 7)
ⴢ
(2x 1) (x 4)(x 1)
x1
Simplifying a complex fraction:
Example 5 (Method I)
1(2)
1(x)
1
2x
1
x
2
x(2)
2(x)
2x
1(x)
2
2
x2
1
x
x
x
1(x)
1
1
2x
x
1
ⴢ
2x
x2
2
Example 6 (Method II)
1
1
1
1
2x ⴢ a b
x
x
2
2x
2
2
2x 4
1
2
1
2x ⴢ a b
x
x
1
1
2x
1
2(x 2)
2
2
Rationalizing the denominator:
Removing a radical from the denominator of a fraction is
called rationalizing the denominator.
Example 7
5
3
2
2y2
3 2
2y
5ⴢ 2
3
3 2
2
2y2 ⴢ 2
2y
3
4y
52
3 3 3
2
2y
3
4y
52
2y
Example 8
8 ⴢ 1 113 1112
8
113 111
1 113 1112 ⴢ 1 113 1112
81 113 1112
13 11
8 1 113 1112
4
2
41 113 1112
5