Slide R-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Slide R-1Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Basic Concepts of Algebra

Chapter R


R.1 The Real-Number System

Identify various kinds of real numbers. Use interval notation to write a set of numbers. Identify the properties of real numbers. Find the absolute value of a real number.


Rational Numbers

Numbers that can be expressed in the form p/q, where p and q are integers and q 0.

Decimal notation for rational numbers either terminates (ends) or repeats.

Examples:

a) 0 b)

c) 9 d)

1

8

7

11


Irrational Numbers

The real numbers that are not rational are irrational numbers.

Decimal notation for irrational numbers neither terminates nor repeats.

Examples:

a) 7.123444443443344…

b) 5


Interval Notation


Examples

Write interval notation for each set and graph the set.a) {x|5 < x < 2}

Solution: {x|5 < x < 2} = (5, 2) ( )

b) {x|4 < x 3}Solution: {x|4 < x 3} = (4, 3]

( ]


Properties of the Real Numbers

Commutative propertya + b = b + a and ab = ba

Associative propertya + (b + c) = (a + b) + c anda(bc) = (ab)c

Additive identity propertya + 0 = 0 + a = a

Additive inverse propertya + a = a + (a) = 0


More Properties

Multiplicative identity property

a • 1 = 1 • a = a Multiplicative inverse property

Distributive property

a(b + c) = ab + ac

1 11 ( 0)a a a

a a


Examples

State the property being illustrated in each sentence.

a) 7(6) = 6(7) Commutative

b) 3d + 3c = 3(d + c) Distributive

c) (3 + y) + x = 3 + (y + x) Associative


Absolute Value

The absolute value of a number a, denoted |a|, is its distance from 0 on the number line.

Example:

Simplify.

|6| = 6

|19| = 19


Distance Between Two Points on the Number Line

For any real numbers a and b, the distance between a and b is |a b|, or equivalently |b a|.

Example:

Find the distance between 4 and 3.

Solution: The distance is |4 3| = |7| = 7,

or |3 (4)| = |3 + 4| = |7| = 7.


R.2 Integer Exponents, Scientific

Notation, and Order of Operations Simplify expressions with integer exponents. Solve problems using scientific notation. Use the rules for order of operations.


Integers as Exponents

When a positive integer is used as an exponent, it indicates the number of times a factor appears in a product.For any positive integer n,

where a is the base and n is the exponent.

Example: 84 = 8 • 8 • 8 • 8

For any nonzero real number a and any integer m,a0 = 1 and .

Example: a) 80 = 1 b)

factors

...n

base n

a a a a a a

1m

m

aa

2 52 5

5 5 2 2

1 1x yx y

y y x x


Properties of Exponents

Product rule

Quotient rule

Power rule

(am)n = amn

Raising a product to a power

(ab)m = ambm

Raising a quotient to a power

m n m na a a

( 0)m

m nn

aa a

a

( 0)m m

m

a ab

b b


Examples – Simplify.

a) r 2 • r 5

r (2 + 5) = r 3

b)

c) (p6)4 = p -24 or 1 p24

d) (3a3)4 = 34(a3)4

= 81a12 or 81 a12

e)

99 4 5

4

36 362

18 18

yy y

y 3 32 2 2 2

4 4

3 6 6 6 6

12 12

6

6 12

21 3

7

3

27

27

a b a b

c c

a b a b

c c

a

b c


Scientific Notation

Use scientific notation to name very large and very small positive numbers and to perform computations.

Scientific notation for a number is an expression of the type N 10m,

where 1 N < 10, N is in decimal notation, and m is an integer.


Examples

Convert to scientific notation.a) 17,432,000 = 1.7432 107

b) 0.00000000024 = 2.4 1010

Convert to decimal notation.a) 3.481 106 = 3,481,000

b) 5.874 105 = 0.00005874


Another Example

Chesapeake Bay Bridge-Tunnel. The 17.6-mile-long tunnel was completed in 1964. Construction costs were $210 million. Find the average cost per mile.

88 1

1

7

2.1 101.19 10

1.76 10

$1.19 10


Rules for Order of Operations

Do all calculations within grouping symbols before operations outside. When nested grouping symbols are present, work from the inside out.

Evaluate all exponential expressions. Do all multiplications and divisions in order from left to

right. Do all additions and subtractions in order from left to

right.


Examples

a) 4(9 6)3 18 = 4 (3)3 18

= 4(27) 18

= 108 18

= 90

b) 3 2

15 (7 2) 20 15 5 20

3 2 27 43 20 23

31 31


R.3 Addition, Subtraction, and

Multiplication of Polynomials Identify the terms, coefficients, and degree of a

polynomial. Add, subtract, and multiply polynomials.


Polynomials

Polynomials are a type of algebraic expression.

Examples: 5y 6t

3 17 4

2x x

9 7k


Polynomials in One Variable

A polynomial in one variable is any expression of the type

where n is a nonnegative integer, an,…, a0 are real numbers, called coefficients, and an 0. The parts of the polynomial separated by plus signs are called terms. The degree of the polynomial is n, the leading coefficient is an, and the constant term is a0. The polynomial is said to be written in descending order, because the exponents decrease from left to right.

1 21 2 1 0... ,n n

n na x a x a x a x a


Examples

Identify the terms.4x7 3x5 + 2x2 9

The terms are: 4x7, 3x5, 2x2, and 9.

Find the degree.a) 7x5 3 5b) x2 + 3x + 4x3 3c) 5 0


Addition and Subtraction

If two terms of an expression have the same variables raised to the same powers, they are called like terms, or similar terms.

Like Terms Unlike Terms

3y2 + 7y2 8c + 5b

4x3 2x3 9w 3y

We add or subtract polynomials by combining like terms.


Examples

Add: (4x4 + 3x2 x) + (3x4 5x2 + 7)

(4x4 + 3x4) + (3x2 5x2) x + 7

(4 + 3)x4 + (3 5)x2 x + 7

x4 2x2 x + 7

Subtract: 8x3y2 5xy (4x3y2 + 2xy)

8x3y2 5xy 4x3y2 2xy

4x3y2 7xy


Multiplication

To multiply two polynomials, we multiply each term of one by each term of the other and add the products.

Example: (3x3y 5x2y + 5y)(4y 6x2y)

3x3y(4y 6x2y) 5x2y(4y 6x2y) + 5y(4y 6x2y)

12x3y2 18x5y2 20x2y2 + 30x4y2 + 20y2 30x2y2

18x5y2 + 30x4y2 + 12x3y2 50x2y2 + 20y2


More Examples

Multiply: (5x 1)(2x + 5)

= 10x2 + 25x 2x 5= 10x2 + 23x 5

Special Products of Binomials

Multiply: (6x 1)2

= (6x)2 + 2• 6x • 1 + (1)2

= 36x2 12x + 1


R.4 Factoring

Factor polynomials by removing a common factor. Factor polynomials by grouping. Factor trinomials of the type x2 + bx + c. Factor trinomials of the type ax2 + bx + c, a 1, using the FOIL method and the grouping method. Factor special products of polynomials.


Terms with Common Factors

When factoring, we should always look first to factor out a factor that is common to all the terms.

Example: 18 + 12x 6x2

= 6 • 3 + 6 • 2x 6 • x2

= 6(3 + 2x x2)


Factoring by Grouping

In some polynomials, pairs of terms have a common binomial factor that can be removed in the process called factoring by grouping.

Example: x3 + 5x2 10x 50

= (x3 + 5x2) + (10x 50)

= x2(x + 5) 10(x + 5)

= (x2 10)(x + 5)


Trinomials of the Type x2 + bx + c

Factor: x2 + 9x + 14.

Solution:

1. Look for a common factor.

2. Find the factors of 14, whose sum is 9.

Pairs of Factors Sum

1, 14 15 2, 7 9 The numbers we need.

3. The factorization is (x + 2)(x + 7).


Another Example

Factor: 2y2 20y + 48.1. First, we look for a common factor.

2(y2 10y + 24)2. Look for two numbers whose product is 24 and whose

sum is 10.Pairs Sum Pairs Sum 1, 24 25 2, 12 14 3, 8 11 4, 6 10

3. Complete the factorization: 2(y 4)(y 6).


Trinomials of the Type ax2 + bx + c, a 1

Method 1: Using FOIL

1. Factor out the largest common factor.

2. Find two First terms whose product is ax2.

3. Find two Last terms whose product is c.

4. Repeat steps (2) and (3) until a combination is found

for which the sum of the Outside and Inside products

is bx.


Example

Factor: 8x2 + 10x + 3.

(8x + )(x + )

(8x + 1)(x + 3) middle terms are wrong 24x + x = 25x

(4x + )(2x + )

(4x + 1)(2x + 3) middle terms are wrong 12x + 2x = 14x

(4x + 3)(2x + 1) Correct! 4x + 6x = 10x


Grouping Method

1. Factor out the largest common factor.

2. Multiply the leading coefficient a and the constant c.

3. Try to factor the product ac so that the sum of the factors is b.

4. Split the middle term. That is, write it as a sum using the factors found in step (3).

5. Factor by grouping.


Example

Factor: 12a3 4a2 16a.1. Factor out the largest common factor, 4a.

4a(3a2 a 4)2. Multiply a and c: (3)(4) = 12.3. Try to factor 12 so that the sum of the factors is the coefficient of

the middle term, 1. (3)(4) = 12 and 3 + (4) = 1

4. Split the middle term using the numbers found in (3). 3a2 + 3a 4a 4

5. Factor by grouping. 3a2 + 3a 4a 4 = (3a2 + 3a) + (4a 4) = 3a(a + 1) 4(a + 1)

= (3a 4)(a + 1)Be sure to include the common factor to get the complete factorization. 4a(3a 4)(a + 1)


Special Factorizations

Difference of Squares

A2 B2 = (A + B)(A B)

Example: x2 25 = (x + 5)(x 5)

Squares of Binomials

A2 + 2AB + B2 = (A + B)2

A2 2AB + B2 = (A B)2

Example: x2 + 12x + 36 = (x + 6)2


More Factorizations

Sum or Difference of Cubes

A3 + B3 = (A + B)(A2 AB + B2)

A3 B3 = (A B)(A2 + AB + B2)

Example: 8y3 + 125 = (2y)3 + (5)3

= (2y + 5)(4y2 10y + 25)


R.5 Rational Expressions

Determine the domain of a rational expression. Simplify rational expressions. Multiply, divide, add, and subtract rational expressions. Simplify complex rational expressions.


Domain of Rational Expressions

The domain of an algebraic expression is the set of all real numbers for which the expression is defined.

Example: Find the domain of .

Solution: To determine the domain, we factor the denominator.

x2 + 3x 4 = (x + 4)(x 1) and set each equal to zero.

x + 4 = 0 x 1 = 0

x = 4 x = 1

The domain is the set of all real numbers except 4 and 1.

2

2

9

3 4

x

x x


Simplifying, Multiplying, and Dividing Rational Expressions

Simplify:

Solution:

Simplify:

Solution:

2

2

1

2 1

x

x x

2

2

1 ( 1)( 1)

2 1 (2 1)( 1)

( 1)

x x x

x x x x

x ( 1)

(2 1) ( 1)

x

x x

1

2 1

x

x

2 2

2 2

9 6 3

12 12

a ab b

a b

2 2 2 2

2 2 2 2

9 6 3 3(3 2 )

12 12 12( )

3(3 )( )

12( )( )

3

a ab b a ab b

a b a b

a b a b

a b a b

(3 ) ( ) a b a b

12 4 ( ) ( ) a b a b

3

4( )

a b

a b


Another Example

Multiply:

Solution:

2

2

2 4

3 3 2

x x

x x x

2

2

2 4 2 ( 2)( 2)

3 3 2 3 ( 2)( 1)

( 2)

x x x x x

x x x x x x

x

( 2)( 2)

( 3) ( 2)

x x

x x

( 1)

( 2)( 2)

( 3)( 1)

x

x x

x x


Adding and Subtracting Rational Expressions

When rational expressions have the same denominator, we can add or subtract the numerators and retain the common denominator. If the denominators are different, we must find equivalent rational expressions that have a common denominator.

To find the least common denominator of rational expressions, factor each denominator and form the product that uses each factor the greatest number of times it occurs in any factorization.


Example

Add:

Solution:

The LCD is (3x + 4)(x 1)(x 2).

2 2

9 2 7

3 2 8 3 4

x

x x x x

9 2 7

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

x

x x x x

2

2

9 2 ( 1) 7 ( 2)

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

9 7 2 7 14

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

(3 4) (3 4)9 16

(3 4)( 2)( 1)

x x x

x x x x x x

x x x

x x x x x x

x xx

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

3 4

( 2)( 1)

x x

x

x x


Complex Rational Expressions

A complex rational expression has rational expressions in its numerator or its denominator or both.

To simplify a complex rational expression: Method 1. Find the LCD of all the denominators

within the complex rational expression. Then multiply by 1 using the LCD as the numerator and the denominator of the expression for 1.

Method 2. First add or subtract, if necessary, to get a single rational expression in the numerator and in the denominator. Then divide by multiplying by the reciprocal of the denominator.


Example: Method 1

Simplify:

The LCD of the four expressions is x2y2.

2 2

1 1

1 1x y

x y

2 2

2 2

2 2

2 2

2 2

2 2 2 22 2

2 2

1 1

1 1

1 1( )

( )

( )1 1( )

x yx y

x yx y

x yx y xy x y xy y x

y x x yx y

x y


Example: Method 2

Simplify:

2 2

1 1

1 1x y

x y

2 2

2 2 2 2 2 2

2 2 2 2

2 2 2 2 2 2

2 2

2 2

1 1 1 1

1 1 1 1

y x

x y x y y xy x

x y x y y x

y x y x

xy xy xyy x y x

x y x y x y

y x x y y x

xy y x x

y

2x

2y2 2 2 2

( )xy y x

y x y x


R.6 Radical Notation and Rational

Exponents Simplify radical expressions. Rationalize denominators or numerators in rational expressions. Convert between exponential and radical notation. Simplify expressions with rational exponents.


Notation

A number c is said to be a square root of a if c2 = a.

nth Root

A number c is said to be an nth root of a if cn = a.

The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index.

n a


Examples

Simplify each of the following:

a) = 7, because 72 = 49.

b) = 7, because 72 = 49 and .

c) because .

d) because (4)3 = 64.

e) is not a real number.

49

49

327 3

125 5

3 64 4

4 25

49 (7) 7 3 3

3

3 3 27

5 5 125


Properties of Radicals

Let a and b be any real numbers or expression for which the given roots exist. For any natural numbers m and n (n 1):

1. If n is even,

2. If n is odd,

3.

4.

5.

n na an na a

n n na b ab

( 0)n

nn

a ab

b b

mn m na a


Examples

a)

b)

c)

d)

e)

f)

g)

h)

2( 7) 7 7

33 ( 7) 7

44 43 7 21

72 36 2 6 2

81 819 3

99

53 5 5327 27 3 243

6 5 6 4

6 4

3 2 2 3

245 49 5

49 5

7 5 7 5

x y x y y

x y y

x y y y x y

22

25 525

yyy


Another Example

Perform the operation.

2 2

3 2 3 2 3 3 3 2 9 6 6 3 3

3 2 ( 9 1) 6 3 3

6 8 6 9

3 8 6


The Pythagorean Theorem

The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse:

a2 + b2 = c2.

b

c a


Example

Juanita paddled her canoe across a river 525 feet wide. A strong current carried her canoe 810 feet downstream as she paddled. Find the distance Juanita actually paddled, to the nearest foot.

Solution: 2 2 2

2 2525 810

275,625 656,100

931,725

965.3 ft

c a b

c

c

c

c

525 ft

810 ft

x


Rationalizing Denominators or Numerators

Rationalizing the denominator (or numerator) is done by multiplying by 1 in such a way as to obtain a perfect nth power.

Example: Rationalize the denominator.

Example: Rationalize

the numerator.

6 6 7 42 42

7 7 7 749

2 2

4 4

4 4

4 4

a b a b a b

a b

a b

a ba b

a b


Rational Exponents

For any real number a and any natural numbers m and n for which exists,

n a

1/

/

//

, and

1

n n

mnm n m n

m nm n

a a

a a a

aa


Examples

Convert to radical notation and, if possible, simplify.

a)

b)

c)

3/ 4 4 311 11

1/ 21/ 2

1 1 19

9 39

4 / 3 4 33

44 / 3 43

27 ( 27) 531441 81, or

27 27 ( 3) 81


More Examples

Convert each to exponential notation.a)

b)

Simplify.a)

b)

6 6 / 55 8 8ab ab

12 4 4 /12 1/ 3x x x

7 / 3 1/ 3/ 3 1/ 3 27( 2) ( 2) ( 2) ( 2) x x x x

8 8 8 813/7 /8 3/ 4 7 /8 3/ 4 8 13 8 5 5x x x x x x x x x


R.7 The Basics of Equation Solving

Solve linear equations. Solve quadratic


Linear and Quadratic Equations

A linear equation in one variable is an equation that is equivalent to one of the form ax + b = 0, where a and b are real numbers and a 0.

A quadratic equation is an equation that is equivalent to one of the form ax2 + bx + c = 0, where a, b, and c are real numbers and a 0.


Equation Solving Principles

For any real numbers a, b, and c, The Addition Principle:

If a = b is true, then a + c = b + c is true. The Multiplication Principle:

If a = b is true, then ac = bc is true. The Principle of Zero Products:

If ab = 0 is true, then a = 0 or b = 0, and if a = 0 or

b = 0, then ab = 0. The Principle of Square Roots:

If x2 = k, then or x k .x k


Example: Solve 2x + 5 = 7 4(x 2)

2 5 7 4( 2)

2 5 7 4 8 Using the distributive property.

2 5 15 4 Combine like terms.

6 5 15 Using the addition principle.

6 10 Using the addi

x x

x x

x x

x

x

tion principle.

10 Using the multiplication principle.

65

Simplifying.3

x

x


Check:

We check the result

in the original equation.

2 5 7 4( 2)

5 52 5 7 4 2

3 3

10 5 65 7 4

3 3 3

10 15 21 14

3 3 3 3

25 21 4

3 3 325 25

3 3

x x


Example: Solve x2 3x = 10

Write the equation with 0 on one side.2

2

3 10

3 10 0

( 2)( 5) 0

2 0 or 5 0

2 or 5

x x

x x

x x

x x

x x


Check: x2 3x = 10

For x = 2 For x = 5

2

2

3 10

( ) 3( ) 10

4 6 10

10

2

10

2

x x

2

2

3 10

( ) 3( ) 10

25 15

5 5

10

10 10

x x


Example: Solve 5x2 25 = 0

We will use the principle of square roots.

2

2

2

5 25 0

5 25

5

5 or 5

x

x

x

x x

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Slide R-1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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