1. College Algebra
Chapter R

2. Sets of Numbers: N W Z Q H R Natural Numbers Whole Numbers Integers
College Algebra R.1
Sets of Numbers: N W Z Q H R
Natural Numbers
Whole Numbers
Integers
Rational Numbers
Irrational Numbers
Real Numbers
1,2,3,…
0,1,2,3,…
…,-2,-1,0,1,2,…
Fractions
Cannot be fractions
Combination of Q and H

3. Set Notation: Braces { } used to group members/elements of a set
College Algebra R.1
Set Notation:
Braces { } used to group members/elements
of a set
Empty or null set designated by empty braces or
The symbol Ø
To show membership in a set use read
“is an element of” or “belongs to”
Other notation:
“is a subset of”
“is not an element of”

4. List the natural numbers less than 6
College Algebra R.1
List the natural numbers less than 6
List the natural numbers less than 1
{1,2,3,4,5}
{ }
Identify each of the following statements as
true or false
True
False; 2.2 is not a whole number

5. Inequality symbols Greater than > Less than <
College Algebra R.1
Inequality symbols
Greater than >
Less than <
Strict vs. Nonstrict inequalities
strict inequalities means the endpoints
are included in the relation aka “less than
or equal to” and “greater than or equal to”

6. Absolute Value of a Real Number
College Algebra R.1
Absolute Value of a Real Number
The absolute value of a real number a, denoted |a|, is the undirected distance between a and 0 on the number line: |a|>0
9 - |7 – 15| = ? 1

7. Definition of Absolute Value:
College Algebra R.1
Definition of Absolute Value:

8. Division and Zero The quotient of Zero and any real number n is zero
College Algebra R.1
Division and Zero
The quotient of Zero and any real number n is zero
Are undefined

9. Square Roots Cube Roots This also means that This also means that
College Algebra R.1
Square Roots
Cube Roots
This also means that
This also means that

10. Order of Operations 1 grouping symbols 2 exponents and roots
College Algebra R.1
Order of Operations
1 grouping symbols
2 exponents and roots
3 division and multiplication
4 subtraction and addition
23,020.89

11. College Algebra R.1
Homework pg 10 (1-100)

12. Algebraic Expressions Terminology
College Algebra R.2
Algebraic Expressions Terminology
Algebraic Term
Constant
Variable Term
Coefficient
Algebraic Expression
A collection of factors
Single nonvariable number
Any term that contains a variable
Constant factor of a variable term
Sum or difference of algebraic terms

13. Decomposition of Rational Terms
College Algebra R.2
Decomposition of Rational Terms
For any rational term

14. Evaluating a Mathematical Expression
College Algebra R.2
Evaluating a Mathematical Expression
1 replace each variable with an open
Parenthesis ( ).
2 substitute the given replacements for each
variable
3 simplify using order of operations

15. Properties of Real Numbers
College Algebra R.2
Properties of Real Numbers
THE COMMUTATIVE PROPERTIES
Given that a and b represent real numbers:
ADDITION: a + b = b + a
Addends can be combined in any order without
changing the sum.
MULTIPLICATION: a*b=b*a
Factors can be multiplied in any order
without changing the product

16. Properties of Real Numbers
College Algebra R.2
Properties of Real Numbers
THE ASSOCIATIVE PROPERTIES
Given that a, b and c represent real numbers:
ADDITION: (a + b) + c = a + (b + c)
Addends can be regrouped.
MULTIPLICATION: (a*b)*c=a*(b*c)
Factors can be regrouped

17. Properties of Real Numbers
College Algebra R.2
Properties of Real Numbers
THE ADDITIVE AND MULTIPLICATIVE IDENTITIES
Given that x is a real number:
x + 0 = x x = x
Zero is the identity for addition
x*1=x *x=x
one is the identity for multiplication

18. Properties of Real Numbers
College Algebra R.2
Properties of Real Numbers
THE ADDITIVE AND MULTIPLICATIVE INVERSES
given that a, b, and x represent real numbers
where a, b = 0
-x + x = 0 x + (-x) = 0
-x is the additive inverse for any real number x
is the multiplicative inverse for any real
number

19. Properties of Real Numbers
College Algebra R.2
Properties of Real Numbers
THE DISTRIBUTIVE PROPERTY OF
MULTIPLICATION OVER ADDITION
Given that a, b, and c represent real numbers:
a(b+c) = ab + ac
A factor outside a sum can be distributed
to each addend in the sum
ab + ac = a(b+c)
A factor common to each addend in a sum can
be “undistributed” and written outside a group

20. College Algebra R.2
Homework pg 19 (1-98)

21. An exponent tells us how many times the base b is used as a factor
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
EXPONENTIAL NOTATION
An exponent tells us how many times the base b
is used as a factor
What would look like?

22. PRODUCT PROPERTY OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT PROPERTY OF EXPONENTS
For any base b and positive integers m and n:
POWER PROPERTY OF EXPONENTS
For any base b and positive integers m and n:

23. PRODUCT TO A POWER PROPERTY
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT TO A POWER PROPERTY
For any bases a and b, and positive integers m, n, and p:
QUOTIENT TO A POWER PROPERTY
For any bases a and b, and positive integers m, n, and p:

24. QUOTIENT PROPERTY OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
QUOTIENT PROPERTY OF EXPONENTS
For any base b and integer exponents m and n:
ZERO EXPONENT PROPERTY
For any base b:

25. PROPERTY OF NEGATIVE EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTY OF NEGATIVE EXPONENTS

26. PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS

27. PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS

28. A number written in scientific notation has the form
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATION
A number written in scientific notation has the form

29. College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATION
Write in scientific notation
Write in standard notation
Simplify and write in scientific notation

30. POLYNOMIAL TERMINOLOGY
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
POLYNOMIAL TERMINOLOGY
Monomial
Degree
Polynomial
Degree of a polynomial
Binomial
Trinomial
A term using only whole number exponents
The same as the exponent on the variable
A monomial or any sum or difference of monomial terms
Is the largest exponent occurring on any variable
A polynomial with two terms
A polynomial with three terms

31. ADDING AND SUBTRACTING POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
Show horizontal and vertical method

32. ADDING AND SUBTRACTING POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
Show horizontal and vertical method

33. PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
Show FOIL and box method

34. PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
Show FOIL and box method

35. PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
Show FOIL and box method

36. SPECIAL POLYNOMIAL PRODUCTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SPECIAL POLYNOMIAL PRODUCTS
Binomial conjugates
For any given binomial, its conjugate is found by using the same two terms
with the opposite sign between them
Product of a binomial and its conjugate
Square of a binomial

37. College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
Homework pg 31 (1-140)

38. Greatest Common Factor
College Algebra R.4 Factoring Polynomials
Greatest Common Factor
Largest factor common to all terms in a polynomial

39. Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping

40. Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping

41. Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials

42. Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials

43. Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Difference of 2 perfect squares
Perfect square trinomials

44. Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms

45. Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Sum or difference of two cubes

46. Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Sum or difference of two cubes

47. U-substitution or placeholder substitution
College Algebra R.4 Factoring Polynomials
U-substitution or placeholder substitution

48. Factoring Polynomials
College Algebra R.4 Factoring Polynomials
Factoring flow chart on page 41
Factoring Polynomials
GCF
Number of Terms
Two
Three
Four
Grouping
Trinomials (a=1)
Difference of Squares
Trinomials (a=1)
Advanced Methods (4.2)
Difference of Cubes
Sum of Cubes

49. College Algebra R.4 Factoring Polynomials
Classwork

50. College Algebra R.4 Factoring Polynomials
Homework pg 41 (1-60)

51. FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
If P, Q, and R are polynomials, where Q or R = 0 then,

52. FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Reduce to lowest terms

53. FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Simplify each and state the excluded values.

54. MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
Factor all numerators and denominators
Reduce common factors
Multiply (numerator x numerator)
(denominator x denominator)

55. MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
Factor all numerators and denominators
Reduce common factors
Multiply (numerator x numerator)
(denominator x denominator)

56. MULTIPLYING RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
Factor all numerators and denominators
Reduce common factors
Multiply (numerator x numerator)
(denominator x denominator)

57. DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS
Invert divisor and multiply as before

58. DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS

59. ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS
Find LCD of all denominators
Build equivalent expressions
Add or subtract numerators
Write the result in lowest terms

60. ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS

61. ADDITION AND SUBTRACTION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS

62. Simplifying Compound Fractions
College Algebra R.5 Rational Expressions
Simplifying Compound Fractions

63. College Algebra R.5 Rational Expressions
Homework pg 50 (1-84)

64. The square root of For any real number The cube root of
College Algebra R.6 Radicals and Rational Exponents
The square root of
For any real number
The cube root of

65. The nth root of For any real number a,
College Algebra R.6 Radicals and Rational Exponents
The nth root of
For any real number a,

66. If is a real number with and , then
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
If is a real number with and , then

67. For m, with m and n relatively prime and
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
For m, with m and n relatively prime and

68. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Product property of radicals

69. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Product property of radicals

70. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Quotient property of radicals

71. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions

72. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions

73. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions

74. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
Rationalizing the denominator

75. PROPERTIES OF RADICALS AND SIMPLIFYING RADICAL EXPRESSIONS
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
Rationalizing the denominator

76. College Algebra R.6 Radicals and Rational Exponents
Homework pg 64 (1-62)

77. Review True False College Algebra R State true or false.
Simplify by combining like terms

78. College Algebra R
Review
Simplify

79. College Algebra R
Review
Simplify

80. College Algebra R
Review
Simplify

81. College Algebra R
Review
Factor

82. College Algebra R
Homework pg