Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition

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Name ___________________________________________________________ Date ____________
Chapter 1 Fundamentals

1.1 Real Numbers

I. Real Numbers

Give examples or descriptions of the types of numbers that make up the real number system.

Natural numbers: aa aa aa aa a a a .

Integers: a a a a aaa aaa a aa aa aa aa aa aa a a a .

Rational numbers: aaaaaaaaaaa aa aaaaaa aaa aaaaaa aa aaaaaaaa .

Irrational numbers: aaaa aaaaaaa aaaa aaaaaa aa aaaaaaaaa aa a aaaaa aa aaaaaaaa .

The set of all real numbers is usually denoted by the symbol .

The corresponding decimal representation of a rational number is aaaaaaaaa . The
corresponding decimal representation of an irrational number is aaaaaaaaaaaa .

II. Properties of Real Numbers

Let a, b, and c be any real numbers. Use a, b, and c to write an example of each of the following properties of
real numbers.

Commutative Property of Addition: a a a a a a a .
Commutative Property of Multiplication: aa a aa .
Associative Property of Addition: aa a aa a a a a a aa a aa .
Associative Property of Multiplication: aaaaa a aaaaa .
Distributive Properties: aaa a aa a aa a aa .
aa a aaa a aa a aa .

Example 1: Use the properties of real numbers to write 4(q+r) without parentheses.

III. Addition and Subtraction

The additive identity is a because, for any real number a, a a a a a . Every real
number a has a aaaaaaaa , −a, that satisfies a(a) a .

To subtract one number from another, simply aaa aaa aaaaaaaa aa aaaa aaaaaa .

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved. 1
2 CHAPTER 1 | Fundamentals
Complete the following Properties of Negatives.

1. (1)a aa .
2. (a) a .
3. (a)b aaaaa a aaaaa .
4. (a)(b) aa .
5. (ab) aa a a .
6. (ab) a a a .

Example 2: Use the properties of real numbers to write 3(2a5b) without parentheses.

IV. Multiplication and Division

The multiplicative identity is a because, for any real number a, a a a a a . Every
nonzero real number a has an aaaaaaa , 1/a, that satisfies a(1/a) a .

To divide by a number, simply aaaaaaaa aa aaa aaaaaaa aa aaaa aaaaaa .

Complete the following Properties of Fractions.

a c
1.   aaaa a aaaa .
b d
a c
2.   aa a aa a aa a aa .
b d
a b
3.   aa a aa a a .
c c
a c
4.   aaa a aaa a aa .
b d
ac
5.  a a a .
bc
a c
6. If  , then aa a aa .
b d

4 19
Example 3: Evaluate: 
9 30
aaaaa

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved.
SECTION 1.1 | Real Numbers 3
V. The Real Line

On the real number line shown below, the point corresponding to the real number 0 is called the
aaaaa . Given any convenient unit of measurement, each positive number x is represented by
aaa aa aaa aaaa a aaaaaaaa aa a aaaaa aa aaa aaaaa aa aaa aaaaaa . Each negative number −x
is represented by aaa aaaaa a aaaaa aa aaa aaaa aa aaa aaaaaa .

0
The real numbers are ordered, meaning that a is less than b, written a a a , if a a aaaaaa
aaaaaa . The symbol ab is read as a aa aaaa aaaa aa aaaaa aa a .

VI. Sets and Intervals

A set is a aaaaaaaaaa aaaaaaa , and these objects are called the aaaaaaaa of
the set. The symbol  means aa aa aaaaaaa aa , and the symbol  means aa aaa
aa aaaaaaa aa .

Name two ways that can be used to describe a set.

aaaa aaaa aaa aa aaaaaaaaa aa aaaaaaa aaaaa aaaaaaaa aaaaaa aaaaaaa aaaaaaa aaa aa aaaaaaaa a aaa aa aa aaa
aaaaaaaaaaa aaaaaaaaa

The union of two sets S and T is the set ST that consists of aaa aaaaaaaa aaaa aaa aa a aa
a aa aa aaaa . The intersection of S and T is the set ST that consists of aaa
aaaaaaaa aaaa aaa aa aaaa a aaa a . The symbol ∅ represents aaa aaaaa
aaaa aaaa aaa aaa aaa aaaa aaaaaaaa aa aaaaaaa .

Example 4: If A={2,4,6,8,10}, B{4,8,12,16} , and C={3,5,7}, find the sets
(a) AB
(b) AB
(c) BC

aaa aaa aa aa aa aaa aaa aaa
aaa aaa aa
aaa a

If a<b, then the open interval from a to b consists of aaa aaaaaaa aaaaaaa a aaa a and
is denoted aaa aa . The closed interval from a to b includes aaaaaaaaaa and
is denoted aaa aa .
Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved.
4 CHAPTER 1 | Fundamentals
VII. Absolute Value and Distance

The absolute value of a number a, denoted by a a a , is aaa aaaaa aa a aa aaa aaaa aaaaaa
aaaa . Distance is always aaaaaaaa aa aaaa , so we have a 0 for
every number a.

If a is a real number, then the absolute value of a is


a 


Example 5: Evaluate.
(a) 128
(b) 915
(c) 77
aaa a
aaa a
aaa a

Complete the following descriptions of properties of absolute value.

1. The absolute value of a number is always aaaaaaaa aa aaaa .
2. A number and its negative have the same aaaaaaaa aaaaa .
3. The absolute value of a product is aaa aaaaaaa aa aaa aaaaaaaa aaaaaa .
4. The absolute value of a quotient is aaa aaaaaaaa aa aaa aaaaaaaa aaaaaa .

If a and b are real numbers, then the distance between the points a and b on the real line is
d(a,b) a a a a a .

Example 6: Find the distance between the numbers −16 and 7.
aa

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Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
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SECTION 1.2 | Exponents and Radicals 5

Name ___________________________________________________________ Date ____________

1.2 Exponents and Radicals

I. Integer Exponents

If a is any real number and n is a positive integer, then the aaa aaaaa aa a is an aa a.
nfactors
The number a is called the aaaa , and n is called the aaaaaaaa .

If a0 is any real number and n is a positive integer, then a0  a and an  aaaa .

Example 1: Evaluate.
0
(a) (2)5 (b) 1 (c) 42
 
9
aaaa aaaaaaaa aaaaaa aaaa

II. Rules for Working with Exponents

Complete the following Laws of Exponents.

1. aman = aaaa .
am
2. = aaaa .
an
3. (am)n = aaa .
4. (ab)n = aaaa .
n
a
5.    aa a aa .
b
n
a
6.    a a a a aa .
b
an
7.  aa a aa .
bm

Example 2: Evaluate.
3
(a) y6y8 (b) (w5)3 (c) b
 
2
aaa aaaaaaaa aaaaaaaa aaaa

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th
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6 CHAPTER 1 | Fundamentals
III. Scientific Notation

Scientists use exponential notation as a compact way of writing aaaa aaaaa aaaaaaa aaa aaaa aaaaa
aaaaaaa .

A positive number x is said to be written in aaaaaaaaaa aaaaaaaa if it is expressed as xa10n,
where 1a10 and n is an integer.

Example 3: Write each number in scientific notation.
(a) 1,750,000 (b) 0.0000000429

aaa aaaa a aaaaaaa aaaa a aaaa

IV. Radicals

The symbol means aaaa aaaaaaaa aaaaaa aaaa aaa .

If n is any positive integer, then the principal nth root of a is defined as follows:
na =b means aa a a .
If n is even, we must have a a a aaa a a a .

Complete the Properties of nth Roots.

1. nab = a .
a
2. n = .
b
3. mna = .
4. nan = a aa a aa aaa .
5. nan = a a a aa a aa aaaa .

Example 4: Evaluate: 464w5y8

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
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SECTION 1.2 | Exponents and Radicals 7
V. Rational Exponents

For any rational exponent m/n in lowest terms, where m and n are integers and n > 0, we define
am/n = or equivalently am/n = nam
If n is even, then we require that a a a .

Example 5: Evaluate.
a) b7/8b9/8
b) 4 x2(x5)2

aa aa
aa aa

VI. Rationalizing the Denominator; Standard Form

Rationalizing the denominator is the procedure in which a aaaaaaa aa a aaaaaaaaa aa aaaaaaaaaa aa
aaaaaaaaaaa aaaa aaaaaaaaa aaa aaaaaaaaaaa aa aa aaaaaaaaaaa aaaaaaaaaa .

Describe a strategy for rationalizing a denominator.

A fractional expression whose denominator contains no radicals is said to be in aaaaaaaa aaaa .

x
Example 6: Rationalize the denominator:
3y

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved.
8 CHAPTER 1 | Fundamentals
Additional notes

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Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
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SECTION 1.3 | Algebraic Expressions 9

Name ___________________________________________________________ Date ____________

1.3 Algebraic Expressions

A variable is a aaaaaa aaaa aaa aaaaaaaa aaa aaaaaa aaaa a aaaaa aaa aa aaaaaaa .

An algebraic expression is aaa aaaaaaaaa aa aaaaaaaa aaaa aa aa aa aaa aa aaa aaaa aaaa
aaaaaaa aaaaa aaaaaaaaa aaaaaaaaaaaa aaaaaaaaaaaaaaa aaaaaaaaa aaaa a aaaaa .

A monomial is aa aaaaaaaaaa aa aaa aaaa aaaa aaaaa a aa a aaaa aaaaaa aaa a aa
a aaaaaaaaaaa aaaaaaa .

A binomial is a aaa aa aaa aaaaaaaaa .

A trinomial is a aaa aa aaaaa aaaaaaaaa .

A polynomial in the variable x is an expression of the form a xna xn1 axa ,
n n1 1 0
where a ,a, ,a are real numbers, and n is a nonnegative integer. If a 0, then the polynomial has
0 1 n n
degree a . The monomials a xk that make up the polynomial are called the
k
aaaaa of the polynomial.

The degree of a polynomial is aaa aaaaaaa aaaaa aa aaa aaaaaaaa aaaa aaaaaaa aa
aaa aaaaaaaaaa .

I. Adding and Subtracting Polynomials

We add and subtract polynomials by aaaaa aaa aaaaaaaaaa aa aaaa aaaaaaa aaaa aaaa aaaaaaaaa
aa aaaaaaa aaa . The idea is to combine aaaa aaaaa , which are terms
with the same variables raised to the same powers, using the aaaaaaaaaa aaaaaaaa .

When subtracting polynomials, remember that if a minus sign precedes an expression in parentheses, then aaa
aaaa aa aaaaa aaaa aaaaaa aaa aaaaaaaaaaa aa aaaaaaa aaaa aa aaaaaa aaa aaaaaaaaaaa .

II. Multiplying Algebraic Expressions

Explain how to find the product of polynomials or other algebraic expressions.

aaa aaa aaaaaaaaaaaa aaaaaaaa aaaaaaaaaaa aa aaaaaaaaaaa aaaaa aa aaaaa aaaaa aa aaa aaaaaaa aa aaa
aaaaaaaaaa aa aaa a aaaa aaaa aaaa aa aaaaaaaa aaa aaa aaaaaaa aa aaaaaaaaaaa aaaa aaaa aa aaa aaaaaa aa aaaa
aaaa aa aaa aaaaa aaaaaa aaa aaaaaa aaaaa aaaaaaaaa

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved.
10 CHAPTER 1 | Fundamentals
Explain the acronym FOIL.

aaa aaaaaaa aaaa aaaaa aa aaaaaaaa aaaa aaa aaaaaaa aa aaa aaaaaaaaa aa aaa aaa aa aaa aaaaaaaa aa aaa aaaaa
aaaaaa aaa aaaaa aaaaaa aaa aaaaa aaaaaa aaa aaa aaaa aaaaaaa

Example 1: Multiply: (x5)(3x7)

III. Special Product Formulas

Complete the following Special Product Formulas.

Sum and Difference of Same Terms
(A + B)(A  B) = aa a aa a
Square of a Sum and Difference
(A + B)2 = aa a aaa a aa a
(A  B)2 = aa a aaa a aa a
Cube of a Sum and Difference
(A + B)3 = aa a aaaa a aaaa a aa a
(A  B)3 = aa a aaaa a aaaa a aa a

The key idea in using these formulas is the Principle of Substitution, which says that aa aaa aaaaaaaaaa
aaa aaaaaaaaa aaaaaaaaaa aaa aaa aaaaaa aa a aaaaaaa .

Example 2: Find the product: (2y5)2.

IV. Factoring Common Factors

Factoring an expression means aaaaa aaa aaaaaaaaaaaa aaaaaaaa aa aaaaaaa aaa aaaaaaa aa aaaaaaaaa
aaaaaaaaa aaaaaaaaaaa aaaa a aaaaaaa aa aaaaaaa aaaaaaaaaaa .

Note Taking Guide for Stewart/Redlin/Watson Precalculus: Mathematics for Calculus, 7th Edition
Copyright © Cengage Learning. All rights reserved.