College Algebra
Chapter R
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
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”
College Algebra R.1
List the natural numbers less than 6
{1,2,3,4,5}
List the natural numbers less than 1
{}
Identify each of the following statements as
true or false
N  W True
{2.2, 2.3, 2.4 2.5}  W
False; 2.2 is not a whole number
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”
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
College Algebra R.1
Definition of Absolute Value:
 x if x  0
x 
 x if x  0
College Algebra R.1
Division and Zero
The quotient of Zero and any real number n is
zero n  0
0
0  n  0 and
n  0 and
n
0
n
Are undefined
College Algebra R.1
Square Roots
Cube Roots
A  B if B  B  A
 A  0
A  B if B  B  B  A
A  R
This also means that
This also means that
A A  A
 A  A
2
3
3
A 3 A 3 A  A
 A  A
3
3
College Algebra R.1
Order of Operations
1
2
3
4
grouping symbols
exponents and roots
division and multiplication
subtraction and addition
1215
 0.075
75001 

12 

23,020.89
College Algebra R.1
Homework pg 10 (1-100)
College Algebra R.2
Algebraic Expressions Terminology
Algebraic Term
A collection of factors
Constant
Single nonvariable number
Variable Term
Any term that contains a variable
Coefficient
Constant factor of a variable term
Algebraic Expression
Sum or difference of algebraic terms
College Algebra R.2
Decomposition of Rational Terms
For any rational term
A
B  0 
B
A 1 A 1
   A
B B 1 B
x3
 2x
7
1
x  3   2x
7
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
2x2  2x  8
2
2   2   8
22  22  8
848
2
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
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
College Algebra R.2
Properties of Real Numbers
THE ADDITIVE AND MULTIPLICATIVE IDENTITIES
Given that x is a real number:
x+0=x
0+x=x
Zero is the identity for addition
x*1=x
1*x=x
one is the identity for multiplication
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
a b
 1
b a
b a
 1
a b
a
b
is the multiplicative inverse for any real
b
number
a
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
College Algebra R.2
Homework pg 19 (1-98)
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
b n  b b
 b 
  b and
b b
 b 
 b  bn
n times
What would x  23 look like?
n times
x  2x  2x  2
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT PROPERTY OF EXPONENTS
For any base b and positive integers m and n:
b b  b
m
n
m n
52  55
x 3  x10
n 4  n 8
POWER PROPERTY OF EXPONENTS
For any base b and positive integers m and n:
b 
m n
b
mn
x 
3 2
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:
a
m
b

n p
 a b
mp
np
3x 
2 3
QUOTIENT TO A POWER PROPERTY
For any bases a and b, and positive integers m, n, and p:
a
 n
b
m
p

a mp
  np
b

  5a

 2b
3



2
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
QUOTIENT PROPERTY OF EXPONENTS
For any base b and integer exponents m and n:
bm
mn

b
, b0
n
b
ZERO EXPONENT PROPERTY
For any base b:
b  1, if b  0
0
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTY OF NEGATIVE EXPONENTS
n
n
b
1
 n
1
b
 2a
 2
 b
3



1
b

n
b
1
2
?
a
 
b
3hk  6h
2 3
2
k
n

b
 
a
3 2
?
n
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS
x9
?
6
x
x
m5 n 2
?
2
mn
3
mn
9 p q 
9 p q 
pq
3 3 s
2
2 s
4
?
a b
 2   ?
 c 
3
12
3
a
4 8
bc
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PROPERTIES OF EXPONENTS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
SCIENTIFIC NOTATION
A number written in scientific notation has the form
N 10
k
1  N  10 and k is an int eger
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
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
POLYNOMIAL TERMINOLOGY
Monomial
A term using only whole number exponents
Degree
The same as the exponent on the variable
Polynomial
A monomial or any sum or difference of monomial terms
Degree of a polynomial
Is the largest exponent occurring on any variable
Binomial
A polynomial with two terms
Trinomial
A polynomial with three terms
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
ADDING AND SUBTRACTING POLYNOMIALS
x
x
 

3
 5x  9  x  3x  2x  8  ?
3
 5x  9  x3  3x 2  2x  8  ?
 
3
2

College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
2x 13x  2  ?
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
PRODUCT OF TWO POLYNOMIALS
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
x7
x7
Product of a binomial and its conjugate
x  7x  7  ?
 A  B A  B  ?
Square of a binomial
x  7 
2
?
 A  B
2
?
College Algebra R.3 Exponents, Polynomials, and Operations on Polynomials
Homework pg 31 (1-140)
College Algebra R.4 Factoring Polynomials
Greatest Common Factor
Largest factor common to all terms in a polynomial
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Common Binomial Factors and Factoring by Grouping
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring quadratic polynomials
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Difference of 2 perfect squares
A  B   A  B A  B
2
2
Perfect square trinomials
A  2 AB  B   A  B
2
2
2
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Sum or difference of two cubes

2

2
A  B   A  B  A  AB  B
3
3
A  B   A  B  A  AB  B
3
3
2

2

College Algebra R.4 Factoring Polynomials
Factoring Special Forms
Sum or difference of two cubes
8x  125
3
27r  64
3
College Algebra R.4 Factoring Polynomials
U-substitution or placeholder substitution
x
2

2


 2x  2 x  2x  3
2
College Algebra R.4 Factoring Polynomials
Factoring flow chart on page 41
Factoring
Polynomials
GCF
Number of
Terms
Two
Three
Difference of
Squares
Trinomials (a=1)
Difference of
Cubes
Sum of Cubes
Trinomials (a=1)
Four
Grouping
Advanced
Methods (4.2)
College Algebra R.4 Factoring Polynomials
Classwork


6  3x 2  2x  4

n  10n  9
m  8m  5
 7r  3r  3
3b  7b  9
u  v2
x  3 y x  3 y 
a  6x c  2d 
8a  3x3c2  k 2 
2

 5 y xyz3  9x 2 z  2z 3  9
College Algebra R.4 Factoring Polynomials
Homework pg 41 (1-60)
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
If P, Q, and R are polynomials, where Q or R = 0 then,
PR P

QR Q
P PR
and

Q QR
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Reduce to lowest terms
College Algebra R.5 Rational Expressions
FUNDAMENTAL PROPERTY OF RATAIONAL EXPRESSIONS
Simplify each and state the excluded values.
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
P R PR
 
Q S QS
1. Factor all numerators and denominators
2. Reduce common factors
3. Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
1.
2.
3.
Factor all numerators and denominators
Reduce common factors
Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
MULTIPLYING RATAIONAL EXPRESSIONS
1.
2.
3.
Factor all numerators and denominators
Reduce common factors
Multiply (numerator x numerator)
(denominator x denominator)
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS
P R P S PS
   
Q S Q R QR
Invert divisor and multiply as before
College Algebra R.5 Rational Expressions
DIVISION OF RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS
1.
2.
3.
4.
Find LCD of all denominators
Build equivalent expressions
Add or subtract numerators
Write the result in lowest terms
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
ADDITION AND SUBTRACTION OF
RATAIONAL EXPRESSIONS
College Algebra R.5 Rational Expressions
Simplifying Compound Fractions
2 3

3m 2  ?
3
1
 2
4m 3m
College Algebra R.5 Rational Expressions
Homework pg 50 (1-84)
College Algebra R.6 Radicals and Rational Exponents
The square root of
2
2
a,
a a
a : a
For any real number
The cube root of a : a
3
For any real number
3
a,
2
3
3
a a
3
College Algebra R.6 Radicals and Rational Exponents
The nth root of
n
n
a : a
n
For any real number a,
n
an  a
n
a  a when a is odd
n
when a is even
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
If
n
R is a real number with n  Z and n  2 , then
n
3
3
8w
?
27
R  n R1  R
1
n
College Algebra R.6 Radicals and Rational Exponents
RATIONAL EXPONENTS
For m, n  Z with m and n relatively prime and n  2
m
n
 R
m
n
n
R 
m
n
R  R
m
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Product property of radicals
n
AB  A  B and
n
n
n
A  B  AB
n
n
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Product property of radicals
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Quotient property of radicals
n
3
A n A
n
B
B
81
?
3
125x
n
and
n
A n A

B
B
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
Rationalizing the denominator
College Algebra R.6 Radicals and Rational Exponents
PROPERTIES OF RADICALS AND
SIMPLIFYING RADICAL EXPRESSIONS
Operations on Radical Expressions
Rationalizing the denominator
College Algebra R.6 Radicals and Rational Exponents
Homework pg 64 (1-62)
College Algebra R
Review
State true or false.
HR
True
2 Q
False
Simplify by combining like terms


4 x  x  2 x  x3  x
2
6x  x
2
College Algebra R
Review
Simplify
 2a  a b 
3 2
2 4 3
4a12b12
1
 8
x z
25x 2  60x  27
4a  6a  a  9
4
2
College Algebra R
Review
Simplify
3x3  13x 2  x  6
m3
3mm  2 
b2
2
9b
College Algebra R
Review
Simplify
r 6
4r 2 r  10
12 6
8 p  4 10p  75
College Algebra R
Review
Factor

9m3q  3q 4  4q 2  2 p 2
62n  9n  3
7 x
2

 3 4 x  5

College Algebra R
Homework pg 68 1-20