1 Real Numbers Polynomials Factoring Polynomials Rational Expressions Integral Exponents Solving Equations Rational Exponents and Radicals Quadratic Equations

1• Real Numbers• Polynomials• Factoring Polynomials• Rational Expressions• Integral Exponents• Solving Equations• Rational Exponents and Radicals• Quadratic Equations• Inequalities and Absolute Value

Fundamentals of Algebra

1.1Real Numbers

OriginOrigin

Positive DirectionPositivePositive DirectionDirectionNegative DirectionNegativeNegative DirectionDirection

2 2 33

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

• We use real numbers everyday to describe various quantities, such as temperature, salary, annual percentage rate, shoe size, grade point average, and so on.

• Some of the symbols we use to represent real numbers are

• To construct the set of real numbers, we start with the set of natural numbers:

N = {1, 2, 3, …}• To this set we can adjoin other numbers, such as the zero, to create

the set of whole numbers:

W = {0, 1, 2, 3, …}• By adjoining the negatives of the natural numbers, we obtain the

set of integers:I = {…, –3, –2, –1, 0, 1, 2, 3, …}

The Set of Real Numbers

3, 17, 2, 0.6666..., 113, 3.9, 0.12875

• Next, we consider the set Q of rational numbers, numbers of the form a/b, where a and b are integers and b ≠ 0.

• Using set notation we write

Q = {a/b | a and b are integers, b ≠ 0}

• Note that I is contained in Q, since each integer may be written in the form a/b, with b = 1.

• Thus, we say that I is a proper subset of Q, which can expressed symbolically as

I Q

• However, Q is not contained in I since fractions such as 1/2 and 23/25 are not integers.

• We can show the relationship of all these sets as follows:

N W I Q


• Finally, we obtain the set of real numbers by adjoining the set of rational numbers to the set of irrational numbers (Ir).

• Irrational numbers are those that cannot be expressed in the form of a/b, where a, b are integers (b ≠ 0).

• Examples of irrational numbers are

and so on.• Thus, the set

R = Q Ir

comprising all rational numbers and irrational numbers is called the

set of real numbers.


32, 3, 6, ,e

• The set of all real numbers consists of the set of rational numbers plus the set of irrational numbers:


Q

I

W

N

Ir

Representing Real Numbers as Decimals

• Every real number can be written as a decimal. • A rational number can be represented as either a repeating or

terminating decimal. – For example, 2/3 is represented by the repeating decimal

0.666666…

which may also be written , where the bar above indicates that the 6 repeats indefinitely.

– The number 1/2 is represented by the terminating decimal

0.5• When an irrational number is represented as a decimal, it neither

terminates nor repeats. For example,

0.6

2 1.41421... 3.14159... an d

• We can represent real numbers geometrically by points on a real number, or coordinate, line:

• Arbitrarily select a point on a straight line to represent the number 0. This point is called the origin.

• If the line is horizontal, then choose a point at a convenient distance to the right of the origin to represent the number 1.

• The distance between the 0 and the 1 determines the scale of the number line.

Origin

Representing Real Numbers in the Number Line

0 1 2 3 4– 4 – 3 – 2 – 1

• We can represent real numbers geometrically by points on a real number, or coordinate, line:

• The point representing each positive real number x lies x units to the right of 0, and the point representing each negative real number x lies – x units to the left of 0.

• Thus, real numbers may be represented by points on a line in such a way that corresponding to each real number there is exactly one point

on the line, and vice versa.

Origin

Positive DirectionPositive DirectionNegative DirectionNegative Direction

p

Representing Real Numbers in the Number Line

2 3

0 1 2 3 4– 4 – 3 – 2 – 1

Operations with Real Numbers

• Two real numbers may be combined to obtain a real number.• The operation of addition, written +, enables us to combine any two

numbers a and b to obtain their sum, denoted a + b.• Another operation, multiplication, written ·, enables us to combine

any two real numbers a and b to form their product, the number a · b (more simply written ab).

Rules of Operation for Real Numbers

Property Example

1. a + b = b + a 2 + 3 = 3 + 2

2. a + (b + c) = (a + b) + c 4 + (2 + 3) = (4 + 2) + 3

3. a + 0 = a 6 + 0 = 6

4. a + (– a) = 0 5 + (– 5) = 0

Properties of Addition

• The operation of subtraction is defined in terms of addition.

• If we let – b be the additive inverse of b, the expression

a + ( – b)may be written in the more familiar form

a – band we say that b is subtracted from a.



• The property of associativity does not apply for subtraction. For Example:

a – (b – c) ≠ (a – b) – c 4 – (2 – 3) ≠ (4 – 2) – 3

• The property of commutativity does not apply for subtraction either. For Example:

a – b ≠ b – a 2 – 3 ≠ 3 – 2

Property Example

1. – (– a) = a – (– 6) = 6

2. (– a)b = (– ab) = a(– b) (– 3)4 = (– 3 · 4) = 3(– 4)

3. (– a)(– b) = ab (– 3)(– 4) = 3 · 4

4. (– 1)a = – a (– 1)5 = – 5

Properties of Negatives


Property Example

1. ab = ba 2 · 3 = 3 · 2

2. a(bc) = (ab)c 4 · (2 · 3) = (4 · 2) · 3

3. a · 1 = a 5 · 1 = 5

4.

Properties of Multiplication

11a

a

13 1

3


• The operation of division is defined in terms of multiplication. • Recall that the multiplicative inverse of a nonzero real number b

is 1/b, also written as b–1.

•

that a is divided by b.

•

• Zero does not have a multiplicative inverse since division by zero is not defined.

1 aa a b

b b

Then, is written or , and we say

1 44

3 3

Thu , =s .


• The property of associativity does not apply for division. For Example:

a ÷ (b ÷ c) ≠ (a ÷ b) ÷ c 4 ÷ (2 ÷ 3) ≠ (4 ÷ 2) ÷ 3

• The property of commutativity does not apply for division either. For Example:

a ÷ b ≠ b ÷ a 2 ÷ 3 ≠ 3 ÷ 2


Property

Example

Properties of Quotients

if ( , 0)a c

ad bc b db d

1.


3 9 because 3 12 9 4

4 12

Property

Example


( , 0)ca a

b ccb b

2.


4 3 3

4 8 8

Property

Example


( 0)a a a

bb b b

3.


4 4 4

3 3 3

Property

Example


( , 0)a c ac

b db d bd

4.


3 5 15

4 2 8

Property

Example


( , , 0)a c a d

b c db d b c

5.


3 5 3 2 3

4 2 4 5 10

Property

Example


( , 0)a c ad bc

b db d bd

6.


3 5 3 2 4 5 13

4 2 8 4

Property

Example


( , 0)a c ad bc

b db d bd

7.


3 5 3 2 4 5 7

4 2 8 4

1.2 Polynomials

Exponents• Expressions such as 25, (–3)2, and (1/4)4 are

exponential expressions.• More generally, if n is a natural number and

a is a real number, then an represents the product of a and itself n times.

Exponents

• If a is a real number and n is a natural number, thenan = a · a · a · a · · · · · a 34 = 3 · 3 · 3 · 3

• The natural number n is called the exponent, and the real number a is called the base.

Exponents• Examples:

44 4 4 4 4 256

31 1 1 1 1

2 2 2 2 8

21 1 1 1

3 3 3 9

35 5 5 5 125

Property 1

• If m and n are natural numbers is a is a real number, then

am · an = am + n 32 · 33 = 32 + 3 = 35

Property 1Examples:

2 3 2 3 53 3 3 3 243

3 1 3 4 43 3 3 3 81x x x x x

2 5 2 5 72 2 2 2 128

Polynomial in One Variable

• A polynomial in x is an expression of the form anxn + an-1xn-1 + + a1x + a0

where n is a nonnegative integer and a0, a1, … , an are real numbers, with an ≠ 0.

• The expressions akxk in the sum are called the terms of the polynomial.

• The numbers a0, a1, a2, … , an are called the coefficients of 1, x, x2, …, xn respectively.

• The coefficient an of xn (the highest power in x) is called the leading coefficient of the polynomial.

• The nonnegative integer n gives the degree of the polynomial.

a

Polynomial in One VariableExample:• Consider the polynomial:

• The terms of the polynomial are –2x5, 8x3, – 6x2, 3x, and 1, respectively.

• The coefficients of 1, x, x2, x3, and x5 are 1, 3, – 6, 8, –2, respectively.

• The leading coefficient of the polynomial is –2.• The degree of the polynomial is 5.

5 3 22 8 6 3 1x x x x

Polynomial in One Variable• A polynomial having just one term is called a

monomial. – For example:

• A polynomial having exactly two terms is called a binomial. – For example:

• A polynomial having exactly three terms is called a trinomial.– For example:

• A polynomial consisting of one constant term a0 is called a constant polynomial.– For example: – 8

38x

47 5x x

6 39 8 2x x x

Polynomial in Several Variables• Most of the terminology used for a polynomial in one variable

is applicable to polynomials in several variables.• But the degree of a term in a polynomial in several variables is

obtained by adding the powers of all variables in the term, and the degree of the polynomial is given by the highest degree of all its terms.

• For example, the polynomial

is a polynomial in the two variables x and y.• It has five terms with degrees 7, 4, 3, 1, and 0, respectively.• Accordingly, the degree of the polynomial is 7.

2 5 3 22 3 8 3 4x y xy xy y

Adding and Subtracting Polynomials• Constant terms and terms having the same variables and

exponents are called like or similar terms.• Like terms may be combined by adding or subtracting their

numerical coefficients.• For example, we can use the distributive property of the real

number system

to perform

and

( )ab ac a b c

3 7 (3 7) 10x x x x

2 2 2 21 1 53 3

2 2 2m m m m

Adding and Subtracting Polynomials• Examples:

3 2 3 2(3 2 4 5) ( 2 2 2)x x x x x

3 2 3 23 2 4 5 2 2 2x x x x x

3 3 2 23 2 2 2 4 5 2x x x x x

3 4 3x x

Remove parentheses

Group like terms together

Combine the terms

Adding and Subtracting Polynomials• Examples:

4 3 4 3 2(2 3 4 6) (3 9 3 )x x x x x x

4 3 4 3 22 3 4 6 3 9 3x x x x x x 4 4 3 3 22 3 3 9 3 4 6x x x x x x

4 3 26 3 4 6x x x x

Remove parentheses

Group like terms together

Combine the terms

Multiplying Polynomials• To find the product of two polynomials, we again use the

distributive property for real numbers.• For example, to compute the product

we use the distributive law

to obtain( )a b c ab ac

3 (4 2)x x

2

3 (4 2) (3 )(4 ) (3 )( 2)

12 6

x x x x x

x x

Multiplying PolynomialsExamples

• Find the product of Solution

(3 5)(2 3)x x

(3 5)(2 3)x x

3 (2 ) (3 )( 3) (5)(2 ) (5)( 3)x x x x

26 9 10 15x x x

3 (2 3) 5(2 3)x x x

26 15x x

Distributive property

Multiply terms

Combine the terms



• Find the product of Solution

2 2(2 3)(2 1)t t t

2 2(2 3)(2 1)t t t

2 2 2 2

2

(2 )(2 ) (2 )( 1) ( )(2 )

( )( 1) (3)(2 ) (3)( 1)

t t t t t

t t

4 2 3 24 2 2 6 3t t t t t

2 2 2 22 (2 1) (2 1) 3(2 1)t t t t t

4 3 24 2 4 3t t t t


Multiply terms

Combine the terms



• Multiply Solution

2(3 )(4 2 )x y x y

2(3 )(4 2 )x y x y

2

2

(3 )(4 ) (3 )( 2 )

( )(4 ) ( )( 2 )

x x x y

y x y y

3 2 212 6 4 2x xy x y y

2 23 (4 2 ) (4 2 )x x y y x y

3 2 212 4 6 2x x y xy y


Multiply terms

Arrange terms in order of descending

powers of x


Special Products

• Here are some commonly used products of polynomials:

Formula

Example

2 2 2( ) 2a b a ab b

2 2 2

2 2

(2 3 ) (2 ) 2(2 )(3 ) (3 )

4 12 9

x y x x y y

x xy y

Special Products


Formula

Example

2 2 2( ) 2a b a ab b

2 2 2

2 2

(4 2 ) (4 ) 2(4 )(2 ) (2 )

16 16 4

x y x x y y

x xy y

Special Products


Formula

Example

2 2( )( )a b a b a b

2 2

2 2

(2 )(2 ) (2 ) ( )

4

x y x y x y

x y

1.3Factoring Polynomials

3 2 3 21 1x x x x x x 3 2 3 21 1x x x x x x

2 ( 1) 1x x x 2( 1) 1x x x

2( 1)( 1)x x 2( 1)( 1)x x

Factor the first Factor the first two termstwo terms

Rearrange Rearrange the termsthe terms

Factor the Factor the common term common term

x x + 1+ 1

Factoring

• Factoring a polynomial is a process of expressing it as a product of two or more polynomials.

• For example, by applying the distributive property we may write

3x2 – x = x(3x – 1)

and we say that x and 3x – 1 are factors of 3x2 – x.

Common Factors

• The first step in factoring a polynomial is to check if it contains any common factors.

• If it does, the common factor of highest degree is factored out.

• For example, the greatest common factor of

is 2a because

• Thus, we can factor out 2a as follows:

22 4 6a x ax a

2 2 2 22 4 6 3 3aa x ax a ax xa a

22 4 6 2 ( 3 3)a x ax a a ax x

Some Important Factoring Formulas

• Having checked for common factors, the next step in factoring a polynomial is to express the polynomial as the product of a constant and/or one or more prime polynomials.

• The following formulas are very useful in this and should therefore be memorized.


Formula

Example

2 2 ( )( )a b a b a b

2 36 ( 6)( 6)x x x


Formula

Example

2 2 ( )( )a b a b a b

2 2 2 28 2 2(4 )

2(2 )(2 )

x y x y

x y x y


Formula

Example

2 2 ( )( )a b a b a b

6 2 3 2

3 3

9 3 ( )

(3 )(3 )

a a

a a


Formula

Example

2 2 2

2 2 2

2 ( )

2 ( )

a ab b a b

a ab b a b

2 28 16 ( 4)x x x


Formula

Example

2 2 2 2

2

4 4 (2 ) 2(2 )( ) ( )

(2 )

x xy y x x y y

x y

2 2 2

2 2 2

2 ( )

2 ( )

a ab b a b

a ab b a b


Formula

Example

3 3 3

2

27 3

( 3)( 3 9)

z z

z z z

3 3 2 2( )( )a b a b a ab b


Formula

Example

3 3 2 2( )( )a b a b a ab b

3 6 3 2 3

2 2 2 4

8 (2 ) ( )

(2 )(4 2 )

x y x y

x y x xy y

Examples

• Factor the expression

Solution2 4 2 2 2

2 2

16 81 (4 ) (9 )

(4 9 )(4 9 )

x y x y

x y x y

2 416 81x y

Examples

• Factor the expression Solution

4 2 2 2 2 2 2

2 2

4 12 9 (2 ) 2(2 )(3 ) (3 )

(2 3 )

w w v v w w v v

w v

4 2 24 12 9w w v v

Examples• Factor the expression Solution 3 6 3 3

3 3 3 3

3 3 3

3 2 2

3 2 2

27 64 (27 64 )

(3 4 )

[3 (4 ) ]

(3 4 )[3 (3)(4 ) (4 ) ]

(3 4 )[3 12 16 ]

x x x x

x x

x x

x x x x

x x x x

3 627 64x x

Trial-and-Error Factorization• The factors of the second-degree polynomial

px2 + qx + r

where p, q, and r are integers, have the form

where ac = p, ad + bc = q, and bd = r.• Since only a limited number of choices are

possible, we use a trial-and-error method to factor polynomials having this form.

( )( )ax b cx d

Example• Factor the expression x2 – 2x – 3Solution• We first observe that, since the coefficient of

x2 is 1, the only possible first-degree terms are

• Next, we observe that the product of the constant terms is (– 3). This gives us the following possible factors:

( 1)( 3) ( 1)( 3)x x x x or

( )( )x x

Example• Factor the expression x2 – 2x – 3

Solution• We have two possible sets of factors:• Now, the coefficient of x in the polynomial is (– 2). • We multiply the coefficients of the inner terms and

the outer terms and add them to see which set of factors yields (– 2):

( 1)( 3) ( 1)( 3)x x x x or

( 1)( 3)x x

Outer terms

Inner terms

( 1)(1) (1)(3) 2

Coefficients of outer terms


(1)(1) (1)( 3) 2 ( 1)( 3)x x

Example• Factor the expression x2 – 2x – 3

Solution• We have two possible sets of factors:

• Now, the coefficient of x in the polynomial is (– 2). • We multiply the coefficients of the inner terms and

the outer terms and add them to see which set of factors yields (– 2):

( 1)( 3) ( 1)( 3)x x x x or

Outer terms

Inner terms



Example• Factor the expression x2 – 2x – 3Solution• We have two possible sets of factors:

• Now, the coefficient of x in the polynomial is (– 2). • We multiply the coefficients of the inner terms and the

outer terms and add them to see which set of factors yields (– 2):

• Thus, we conclude that the correct factorization is

( 1)( 3) ( 1)( 3)x x x x or

(1)(1) (1)( 3) 2 ( 1)( 3)x x

2 2 3 ( 1)( 3)x x x x

Examples• Use trial and error to factorize the

following expressions:23 4 4x x (3 2)( 2)x x

23 6 24x x 23( 2 8)

3( 4)( 2)

x x

x x

Factoring by Regrouping

• Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out.

Examples• Factor the expressionSolution

3 2 1x x x

3 2 3 21 1x x x x x x

2( 1) 1x x x

2( 1)( 1)x x

Factor the first two terms

Rearrange the terms

Factor the common term

x + 1

Factoring by Regrouping

• Sometimes a polynomial may be factored by regrouping and rearranging terms so that a common term can be factored out.

Examples• Factor the expressionSolution

2 2ax ay bx by

2 2 2 ( ) ( )ax ay bx by a x y b x y

( )(2 )x y a b

Factor 2a from the first two terms and b from the second two terms

Factor the common term x + y

1.4Rational Expressions

2

2

2 8 4 4

2 16

x x x

x x

2

2

2 8 4 4

2 16

x x x

x x

22( 4) ( 2)

2 ( 4)( 4)

x x

x x x

22( 4) ( 2)

2 ( 4)( 4)

x x

x x x

2( 4)( 2)( 2)

( 2)( 4)( 4)

x x x

x x x

2( 4)( 2)( 2)

( 2)( 4)( 4)

x x x

x x x

2( 2)

( 4)

x

x

2( 2)

( 4)

x

x

Rational Expressions

• Quotients of polynomials are called rational expressions.• Examples

• Because division by zero is not allowed, the denominator of a rational expression must not equal zero.

• Thus, in the first example, x ≠ – 3/2, and in the second example, y ≠ 4x.

6 1

2 3

x

x

2 33 2

4

x y xy

x y

Simplifying Rational Expressions

• A rational expression is simplified, or reduced to lowest terms, if its numerator and denominator have no common factors other than 1 and –1.

• If a rational expression does contain common factors, we use the properties of the real number system to write

• This process if often called “canceling common factors.”• To indicate this process, we often write a slash through the common

factors being cancelled:

1ac a c a a

bc b c b b

(a, b, and c are real numbers, and bc ≠ 0)

ac a

bc b

Examples

• Simplify the expression

Solution

2

2

2 3

4 3

x x

x x

2

2

2 3 ( 3)( 1)

4 3 ( 3)( 1)

x x x x

x x x x

( 3)( 1)

( 3)( 1)

x x

x x

( 1)

( 1)

x

x

Factorize numerator and denominator

Cancel any common factors

Examples• Simplify the expression

Solution

23 4 4

2 1

x x

x

23 4 4 (1 2 )(3 2 )

2 1 2 1

x x x x

x x

(2 1)(2 3)

2 1

x x

x

(2 3)x

Factorize numerator and denominator

Rewrite the term 1 – 2x in the form – (2x – 1)

(2 1)(2 3)

2 1

x x

x

Cancel any common factors

Multiplication and Division• If P, Q, R, and S are polynomials, then

Multiplication

Example

P R PR

Q S QS

2 ( 1) 2 ( 1)

( 1) ( 1)

x x x x

y y y y

(Q, S ≠ 0)

Multiplication and Division• If P, Q, R, and S are polynomials, then

Division

Example

P R P S PS

Q S Q R QR

2 2 2 2

2 2

3 1 3 ( 3)

1 ( 1)

x y x x x x

y x y y y y

(Q, R, S ≠ 0)

Examples• Perform the indicated operation and simplify:

2

2

2 8 4 4

2 16

x x x

x x

22( 4) ( 2)

2 ( 4)( 4)

x x

x x x

2( 4)( 2)( 2)

( 2)( 4)( 4)

x x x

x x x

2( 2)

( 4)

x

x


2 2

2

6 9 9

3 12 6 18

x x x

x x x

2 2

2

6 9 6 18

3 12 9

x x x x

x x

2( 3) 6 ( 3)

3( 4) ( 3)( 3)

x x x

x x x

( 3)( 3)(6 )( 3)

3( 4)( 3)( 3)

x x x x

x x x

2 ( 3)

( 4)

x x

x

2

Addition and Subtraction• If P, Q, R, and S are polynomials, then

Addition

Example

P Q P Q

R R R

2 6 2 6 8

2 2 2 2

x x x x x

x x x x

(R ≠ 0)

Addition and Subtraction• If P, Q, R, and S are polynomials, then

Subtraction

Example

P Q P Q

R R R

3 3 2y y y y y

y x y x y x y x

(R ≠ 0)


3 4 4 2

4 3

x y

x y

3 4 3 4 2 4

4 3 3 4

x y y x

x y y x

9 12 16 8

12 12

xy y xy x

xy xy

25 8 12

12

xy x y

xy


1 1

x h x

1 1x x h

x h x x x h

( ) ( )

x x h

x x h x x h

( )

h

x x h

Compound Fractions• A fractional expression that contains

fractions in its numerator or denominator is called a compound fraction.

• The techniques used to simplify rational expressions may be used to simplify these fractions.

Examples• Simplify the expression:

11

14

x

xx

1 11

1 14

xx x

xx

x x

2

1 114

xx

xx

2

2

1 4

x x

x x

2

1 ( 2)( 2)

x x

x x x

( 1)( 2)

x

x x

Examples• Simplify the expression:

2 2

1 1

1 1x y

x y

2 2

2 2

y xxy

y xx y

2 2

2 2

y x x y

xy y x

( )( )

y x x xy y

xy y x y x

( )

xy

y x

1.5Integral Exponents

If If m m and and nn are are integersintegers and and aa is a is a real numberreal number, then, then

1.1. aamm ·· aann = = aam m ++ nn 3322 ·· 3333 = 3= 32 + 32 + 3 = 3= 35 5

2.2.

3.3. ((aamm))nn = = aamnmn ((xx44))33 = = xx44··33 = = xx1212

4.4. ((abab))nn = a= ann ·· bbnn (2(2xx))44 = = 2244xx44 = = 1616xx44

5.5.

mm n

n

aa

a

mm n

n

aa

a

77 4 3

4

xx x

x

77 4 3

4

xx x

x

( 0)n n

n

a ab

b b

( 0)n n

n

a ab

b b

3 3 3

32 2 8

x x x

=3 3 3

32 2 8

x x x

=

Exponents

• Recall that if a is a real number and n is a natural number, then

an = a · a · a · a · · · · · a

• The natural number n is called the exponent, and the real number a is called the base.

n factors

Examples• Write each of the numbers below

without using exponents:52 2 2 2 2 2 32

32

3

2 2 2 8

3 3 3 27

Zero Exponent• For any nonzero real number a,

a0 = 1

• The expression 00 is not defined.

02 1 0( 2) 1 0( ) 1 0

11

3

Examples:

Exponential Expressions With Negative Exponents

• If a is any nonzero real number and n is a positive integer, then

1nn

aa

Examples

• Write each of the numbers below without using exponents:

242

1 1

4 16

32 3

1 1

2 8

12

3

1 22

33

1 1 3

2

33

2

3

3 332

1 2 8

3 27

Properties of Exponents

• If m and n are integers and a is a real number, then

1. am · an = am + n 32 · 33 = 32 + 3 = 35

2.

3. (am)n = amn (x4)3 = x4·3 = x12

4. (ab)n = an · bn (2x)4 = 24x4 = 16x4

5.

mm n

n

aa

a

77 4 3

4

xx x

x

( 0)n n

n

a ab

b b

3 3 3

32 2 8

x x x

=

Examples

• Simplify the expression, writing your answer using positive exponents only:

3 5(2 )(3 )x x 3 5 86 6x x

5

4

2

3

x

x5 42 2

3 3x x

31 32u v9

3 ( 1)(3) 3(3) 3 93

82 8

vu v u v

u

13 4

5 3

2m n

m n

23 5 4 3 1 2 1

2

1(2 ) (2 )

2 2

mm n m n

m n n

1.6Solving Equations

2 1 11

3 4

k k

2 1 11

3 4

k k

2 1 112 12(1)

3 4

k k

2 1 112 12(1)

3 4

k k

2 1 112 12 12

3 4

k k 2 1 1

12 12 123 4

k k

4(2 1) 3( 1) 12k k 4(2 1) 3( 1) 12k k

8 4 3 3 12k k 8 4 3 3 12k k

5 7 12k 5 7 12k

5 5k 5 5k

5 5

5 5

k

5 5

5 5

k

77 12 75k 77 12 75k

1k 1k

Equations• An equation is a statement that two

mathematical expressions are equal.• The following are examples of equations:

2 3 7x y

3(2 3) 4( 1) 4x x

3 1

2 3 4

y y

y y

Equality Properties of Real Numbers

• Let a, b, and c be real numbers.1. If a = b, then

a + c = b + c Addition property and

a – c = b – c Subtraction property 2. If a = b, and c ≠ 0, then

ac = bc Multiplication property and

a b

c c Division property

Linear Equations• A linear equation in the variable x is an equation that can

be written in the form ax + b = 0

where a and b are constants with a ≠ 0.• A linear equation in x is also called a first degree equation

in x or an equation of degree 1 in x.

8 3 2 9x x

28 23 2 9xx xx

6 3 9x

36 33 9x

6 12x

6 2

6 6

1x

2x

Examples

• Use the equality properties of real numbers to solve the equation

Examples• Use the equality properties of real numbers to solve the equation

3 2( 1) 2 4p p p

3 2 2 2 4p p p

5 2 2 4p p

4 25 2 22 pp pp

7 2 4p

27 2 4 2p

7 2

7 7

p

2

7p

7 2p

Examples• Use the equality properties of real numbers to solve the equation

2 1 11

3 4

k k

2 1 1112 12( )

3 4

k k

2 1 112 12 12

3 4

k k

4(2 1) 3( 1) 12k k

8 4 3 3 12k k

5 7 12k

5 5k

5 5

5 5

k

77 12 75k

1k

Examples


2 1

3( 1) 2( 1) 3

x

x x

6( 1) 6( 12 1

3( 1) 2) 6( 1)

( 1) 3

xx

x xx x

4 3 2( 1)x x

4 3 2 2x x

4 5 2x

44 5 2 4x

24 3 22 2xx xx

5 2x

5 2

5 5

x

2

5x

Examples


2 5 3x

222 5 3x

2 5 9x

52 55 9x

2 4x

2 4

2 2

x

2x

1.7Rational Exponents and Radicals

5/3( 8) 5/3( 8)

2/31

27

2/31

27

1/3(64)1/3(64)

3/4(81)3/4(81)

3 64 4 3 64 4

1/4 3 3(81 ) 3 27 1/4 3 3(81 ) 3 27

1/3 5 5( 8 ) ( 2) 32 1/3 5 5( 8 ) ( 2) 32

21/3 21 1 1

27 3 9

21/3 21 1 1

27 3 9

nth Root of a Real Number

• If n is a natural number and a and b are real numbers such that

then we say that a is the nth root of b.

na b


• For n = 2 and n = 3, the roots are commonly referred to as the square roots and the cube roots, respectively.

Examples:• – 2 and 2 are square roots of 4 because (– 2)2 = 4

and 22 = 4.• – 3 and 3 are square roots of 9 because (– 3)2 = 9

and 32 = 9.• – 4 and 4 are square roots of 16 because (– 4)2 = 16

and 42 = 16.


• How many real roots does a real number b have?1. When n is even, the real nth roots of a positive real

number b must come in pairs: one positive and one negative.• For example, the real fourth roots of 81

include – 3 and 3.• To avoid ambiguity we define the principal nth root

of a positive number when n is even to be the positive root


• How many real roots does a real number b have?2. When n is even and b is a negative real number, there

are no real nth roots of b.• For example, if b = – 9 and the real number a is a

square root of b, then by definition a2 = – 9.• But this is a contradiction since the square of a real

number cannot be negative, so b has no real roots in this case.


• How many real roots does a real number b have?3. When n is odd, then there is only one real nth root of b.

• For example, the cube root of – 64 is – 4.

Radicals• We use the notation called a radical, to denote the principal nth

root of b.• The symbol is called a radical sign, and the number b within the

radical sign is called the radicand.• The positive integer n is called the index of the radical.• For square roots (n = 2), we write instead of

n b

b 2 b .

Examples

• Determine the number of roots of the real numberSolution• Here b > 0, n is even, and there is one principal root.• Thus,

25

25 5

Examples

• Determine the number of roots of the real numberSolution• Here b = 0, n is odd, and there is one root.• Thus,

5 0

5 0 0

Examples

• Determine the number of roots of the real numberSolution• Here b < 0, n is odd, and there is one root.• Thus,

3 27

3 27 3

Examples

• Determine the number of roots of the real numberSolution• Here b < 0, n is even, and so no real root exists.• Thus, is not defined.

27

27

Rational Exponents

1. If n is a natural number and b is a real number, then

(If b < 0 and n is even, b1/n is not defined)

1/n nb b

Rational Exponents

2. If m/n is a rational number reduced to its lowest terms (m, n natural numbers), then

or, equivalently,

whenever it exists.

/ 1/( )m n n mb b

/m n mnb b

Examples

• Simplify the expressions:

2/3(27)

2/3( 27)

1/29

1/3( 8)

9 3

3 8 2

1/3 2 2

2 1/3 1/3

(27 ) 3 9

(27 ) (729) 9

1/3 2 2

2 1/3 1/3

( 27 ) ( 3) 9

[( 27) ] 729 9

Examples

• Simplify the expressions:

5/3( 8)

2/31

27

1/3(64)

3/4(81)

3 64 4

1/4 3 3(81 ) 3 27

1/3 5 5( 8 ) ( 2) 32

21/3 21 1 1

27 3 9

Negative Exponents

• If m/n is a rational number reduced to its lowest terms (m, n natural numbers), then

//

1( 0)m n

m na a

a

Examples

• Simplify the expressions:5/24

1/3( 8)

5/2 1/2 5 5

1 1 1 1

4 (4 ) 2 32

1/3

1 1 1

( 8) 2 2

Properties of Radicals

• If m and n are natural numbers and a is a real number for which the indicated roots exist, then

Property

Example

( )nn a a

3 1/3 3 13( 2) (2 ) 2 2



Property

Example

3 3 3 3216 27 8 27 8 3 2 6

n n nab a b



Property

Example

n

nn

a a

b b

3

33

8 8 2 1

64 4 264



Property

Example

m n mna a

3 3 2 664 64 64 2



Property

Example

nnn a aIf is eve n:

2( 3) 3 3



Property

Example

nnn a aIf is odd:

33 ( 2) 2

Simplifying Radicals

• An expression involving radicals is simplified if the following conditions are satisfied:1. The powers of all factors under the radical sign

are less than the index of the radical.2. The index of the radical has been reduced as far

as possible.3. No radical appears in a denominator.4. No fraction appears within a radical.

Examples• Simplify the radical:

3 375 3

333

3

3 125

3 5

5 3


3 6 93 8x y z 3 2 3 33

3 2 3 33 3

2 3

2 ( )

2 ( )

2

xy z

xy z

xy z


4 26 81x y 2 26

2 23 2

23

(9 )

(9 )

9

x y

x y

x y

Rationalizing the Denominator• The process of eliminating a radical from the

denominator of an algebraic expression is referred to as rationalizing the denominator.

• This can be done by multiplying both the numerator and the denominator by the radical that we wish to eliminate.

• For example:

5

5

3 3 3 5 3 5

55 5 25

Examples• Rationalize the denominator:

1

2

1

2

2

4

2

2

21

22

2


3

2

x

x

3

2

3

23

2

x

x

x x

x

x

x

x


3

x

y

23

3

23

33

23

23

x

y

x

y

y

y

y

x y

y

1.8Quadratic Equations

3 41

2

22 3 3 4(1)( 8)4

2 2(1)

b b acx

a

Quadratic Equations• A quadratic equation in the variable x is any

equation that can be written in the form

where a, b, and c are constants and a ≠ 0.• We refer to this form as the standard form.• Examples of quadratic equations in

standard form:

2 0ax bx c

2

2

3 4 2 0

2 8 5 0

x x

x x

Solving by Factoring• We solve a quadratic equation in x by finding its roots.• The roots of a quadratic equation in x are the values

of x that satisfy the equation.• The method of solving quadratic equations by

factoring relies on the following zero-product property of real numbers:

Zero-Product Property of Real Numbers If a and b are real numbers and ab = 0, then

a = 0 , or b = 0, or both a, b = 0.

Examples• Solve by factoring.

x2 – 3x + 2 = 0Solution• Factoring the equation, we find that

x2 – 3x + 2 = 0(x – 2)(x – 1) = 0

• By the zero-product property of real numbers, we have

x – 2 = 0 or x – 1 = 0

from which we see that x = 2 or x = 1 are the roots of the equation.


2x2 – 7x = – 6Solution• Rewriting the equation in standard form, we have

2x2 – 7x + 6 = 0• Factoring the equation, we find that

2x2 – 7x + 6 = 0 (2x – 3)(x – 2) = 0• By the zero-product property of real numbers, we have

2x – 3 = 0 or x – 2 = 0 from which we see that the roots of the equation are

x = 3/2 or x = 2.


4x2 – 3x = 0Solution• Factoring the equation, we find that

4x2 – 3x = 0x(4x – 3) = 0

• By the zero-product property of real numbers, we have

x = 0 or 4x – 3 = 0

from which we see that the roots of the equation are

x = 0 or x = 3/4

Solving by Completing the Square

1. Write the equation ax2 + bx + c = 0 in the form

where the coefficient of x2 is 1 and the constant term is on the right side of the equation.

2 b cx x

a a

2

2

2 2 0

2 2

x x

x x

Example


2. Square half of the coefficient of x.

Example

2/

2

b a

22

12


3. Add the number obtained in step 2 to both sides of the equation, factor, and solve for x.

Example2

2

2 2

( 1) 3

1

3

1

1

1

3

x x

x

x

x

Examples• Solve by completing the square:

Solution1. First write

2. Square half of the coefficient of x, obtaining2

23

3 942 8 64

24 3 2 0x x

2

2

3 10

4 23 1

4 2

x x

x x


Solution3. Add 9/64 to both sides of the equation:

Factoring, we have2

3 41

8 64

3 41

8 8

3 41 1(3 41)

8 8 8

x

x

x

24 3 2 0x x

2 9 9

64 64

3 1

4 2x x


Solution1. First write

2. The coefficient of x is 0, so we can skip

step 2.3. Taking the square root in both sides, we

have9 3 3 2

2 22x

26 27 0x

2 2 96 27

2x x o r

The Quadratic Formula

• The solutions of ax2 + bx + c = 0 (a ≠ 0) are given by

2 4

2

b b acx

a

Examples• Use the quadratic formula to solve

Solution• The equation is in standard form, with a = 2,

b = 5, and c = – 12.

22 5 12 0x x

22 5 5 4(2)( 12)4

2 2(2)

b b acx

a

5 121 5 11

4 43

42

or

Examples• Use the quadratic formula to solve

Solution• We first rewrite the equation in standard form

from which we see that a = 1, b = 3, and c = – 8.

• Thus, or

2 3 8x x

22 3 3 4(1)( 8)4

2 2(1)

b b acx

a

3 41

2

2 3 8 0x x

3 41 3 411.7 4.7

2 2x x

1.9Inequalities and Absolute Value

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

((xx –– 3) 3) –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– 0 +0 + ++ ++ + + + + + + + + + + Sign ofSign of

((xx –– 2)2) –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– –– 0 + + + + + 0 + + + + + ())

Intervals• We described the system of real

numbers in Section 1.1.• We will now be interested in certain

subsets of real numbers called finite intervals and infinite intervals.

• Finite intervals can be open, closed, or half-open.

• Open intervals– The set of all real numbers that lie strictly between two fixed

numbers a and b is called an open interval (a, b).– It consists of all real numbers x that satisfy the inequalities a < x < b.– It is called “open” because neither of its endpoints is included in

the interval.– For example, the open interval (–2, 1) includes all the real

numbers between –2 and 1, but does not include the numbers –2 and 1 themselves.

– Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Finite Intervals

( ))

• Closed intervals– The set of all real numbers between two fixed numbers a and b

that includes the numbers a and b is called a closed interval [a, b].– It consists of all real numbers x that satisfy the inequalities a x b.– It is called “closed” because both of its endpoints are included

in the interval.– For example, the closed interval [–1, 2] includes all the real

numbers between –2 and 1, including the numbers –1 and 2 themselves.

– Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Finite Intervals

[ ]]

• Half-open intervals– The set of all real numbers between two fixed numbers a and b

that includes only one of the endpoint numbers a or b is called a half-open or half-closed interval.

– The half-open interval (a, b] consist of all real numbers x that satisfy the inequalities a < x b.

– For example, the half-open (–3, 1/2] includes all the real numbers between –3 and 1/2, including the number 1/2 but not including the number –3.

– Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Finite Intervals

( ]]

• Half-open intervals– The set of all real numbers between two fixed numbers a and b

that includes only one of the endpoint numbers a or b is called a half-open or half-closed interval.

– The half-open interval [a, b) consist of all real numbers x that satisfy the inequalities a x < b.

– For example, the half-open [–2, 1) includes all the real numbers between –2 and 1, including the number –2 but not including the number 1.

– Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Finite Intervals

Infinite Intervals

• Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a, x a, x < a, and x a, respectively.

• For example, the infinite interval (2, ∞) satisfies x > 2 and includes all the real numbers greater than 2, but does not include the number 2 itself.

• Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11(

• Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy

• x > a, x a, x < a, and x a, respectively.• For example, the infinite interval [2, ∞) satisfies x 2 and

includes all the real numbers greater than 2, including the number 2 itself.

• Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Infinite Intervals

[

• Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy

x > a, x a, x < a, and x a, respectively.• For example, the infinite interval (– ∞, 1) satisfies x < 1 and

includes all the real numbers less than 1, but does not include the number 1 itself.

• Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Infinite Intervals

)

• Infinite intervals include the half lines (a, ∞), [a, ∞), (– ∞, a), and (– ∞, a], defined by the set of real numbers x that satisfy x > a,

x a, x < a, and x a, respectively.• For example, the infinite interval (– ∞, 1] satisfies x 1 and

includes all the real numbers less than 1, including the number 1 itself.

• Graphically:

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

Infinite Intervals

]

Properties of Inequalities

• Let a, b, and c be any real numbers,1. If a < b and b < c, then a < c.

Example

2 < 3 and 3 < 8, so 2 < 8.


• Let a, b, and c be any real numbers,2. If a < b, then a + c < b + c.

Example

Consider –5 < –3 so –5 + 2 < –3 + 2

that is, –3 < –1


• Let a, b, and c be any real numbers,3. If a < b and c > 0, then ac < bc.

Example

Consider –5 < –3 and 2 > 0 so (–5)(2)< (–3)(2)

that is, –10 < –6


• Let a, b, and c be any real numbers,4. If a < b and c < 0, then ac > bc.

Example

Consider –5 < –3 and –2 < 0 so (–5)(–2) > (–3)(–2)

that is, 10 > 6

Examples• Solve the inequality Solution

• The solution is the set of all values of x in the interval (– ∞, 3).

3 2 7x

23 22 7x

3 9x

3 33 9

1 1x

3x

Examples• Solve the inequality Solution

• The solution is the set of all values of x in the interval [2, 6).

1 2 5 7x

1 25 5755x

4 2 12x

4 21

22

1

21

1

2x

2 6x

Solving Inequalities by Factoring

• The method of factoring can be used to solve inequalities that involve polynomials of degree 2 or higher.

• This method relies on the principle that a polynomial changes sign only at a point where its value is 0.

• To find the values where a polynomial is equal to 0, we set the polynomial equal to 0 and then solve for x.

• The values obtained can then be used to help us solve the inequality.

• Solve the inequality Solution• First, set to 0 the polynomial in the inequality and factor the

polynomial:

• Thus, the polynomial changes signs at x = 2 and at x = 3.

Examples2 5 6 0x x

2 5 6 0x x ( 3)( 2) 0x x

• Solve the inequality Solution• Next, we construct a sign diagram for the factors of the polynomial:

• Since x2 – 5x + 6 > 0, we require that the product of the two factors be positive, which occurs when both factors have the same sign.

• The diagram shows us that the two factors have the same sign when x < 2 or x > 3.

• Thus, the solution set is (–∞, 2) (3, ∞).

00 11 22 3 3 44–– 44 –– 33 –– 22 –– 11

(x – 3) – – – – – – – – – – – – – – – – – – – – – – 0 + + + + + + + + Sign of

(x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – 0 + + + + +

Examples2 5 6 0x x

()

• Solve the inequality Solution• First, set to 0 the polynomial in the inequality

and factor the polynomial:

• Thus, the polynomial changes signs at x = – 4 and at x = 2.

Examples2 2 8 0x x

2 2 8 0x x

( 4)( 2) 0x x

• Solve the inequality Solution• Next, we construct a sign diagram for the factors of the polynomial:

• Since x2 + 2x – 8 < 0, we require that the product of the two factors be negative, which occurs when the two factors have the different sign.

• The diagram shows us that the two factors have different signs when – 4 < x < 2.

• Thus, the solution set is (– 4, 2).

Examples2 2 8 0x x

(x + 4) – – – – – – – 0 + + + + + + + + + + + + + + + + + + + + Sign of

(x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – – 0 + + +

( )

• Solve the inequality Solution• First, rewrite the inequality so that the right side equals 0:

• Thus, the quotient changes signs at x = – 1 and at x = 2.

Examples2 1

12

x

x

2 11 0

22 1 ( 2)

02

10

2

x

xx x

xx

x

• Solve the inequality Solution• Next, we construct a sign diagram for the factors of the quotient:

• Since the quotient of these factors must be positive or equal to 0, we require that these two factors have the same sign.

• The diagram shows us that the two factors have same sign for all values in (–∞, –1] (2, ∞).

• Note that x = 2 is not included, since division by 0 is not allowed.

Examples

(x + 1) – – – – – – – – – – – – – – – – 0 + + + + + + + + + + + + + +Sign of

–– 55 –– 44 –– 33 –– 22 –– 11 00 11 22 33

(x – 2) – – – – – – – – – – – – – – – – – – – – – – – – – – 0+ + + + +

(]

2 11

2

x

x

Absolute Value

• The absolute value of a number a is denoted by |a| and is defined by

• |a| =

• Example |5| = 5 and |-5| = 5• Geometrically |a| is the distance between the origin and the

point on the number line that represents the number a

0 if a-

0 if a

{

End of Chapter

Documents

1 Real Numbers Polynomials Factoring Polynomials Rational Expressions Integral Exponents Solving Equations Rational Exponents and Radicals Quadratic Equations