Download Sullivan College Algebra: Section R.1

College Algebra
Math 123-01
Chapter R
Review Elem. College Algebra
Instructor: Dr. Chekad Sarami
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Course Number & Name:
MATH 123 College Algebra
Semester Hours of Credit:
3
Days/Time Class Meets: MWF
Room/Bldg. SBE 108
Instructors Name: Dr. Chekad Sarami
Office Location: SBE 334
Office Telephone:
672-1129
E-mail address: [email protected]
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Office Hours: M2-3 pm, TR 2:00-3:30, F 2-3 pm and 4-5
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pm
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Final : Wednesday, December 7, Time: 6-8pm
COURSE DESCRIPTION:
Mathematics 123 is a college level algebra
course containing topics as follows: Sets,
the real number system, exponents, radicals,
polynomials, equations, inequalities,
relations and functions, graphing,
exponential and logarithmic functions.
Prerequisites: High School Algebra I, II,
and Plane Geometry or equivalent, and
satisfactory placement score.
 Graphical Calculator is Required!

GRADING SCALE:
HW and Three Take-Home Tests (lowest
chapter test score is dropped)
65%
 Group Projects:
10%
 Instructor Option:
5%
 Final Exam (Cumulative)
20%
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Total: 100
Student Tips
I. Prior to the beginning of the
semester:

Begin each course with a positive attitude and open mind.

Determine why you are taking this course; graduation requirement, prerequisite for
another course, your job demands this course, to improve your GPA. These are allimportant reasons—remind yourself of this throughout the course.

Plan your school, work, and recreation schedule to allow plenty of long blocks of time
to study.

Plan or designate a study area that is quiet and where you won’t be interrupted.
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Purchase your textbook, calculator and any supplies before the semester begins or on the
first day of class.

II. During the semester:

Make an exaggerated effort the first couple of weeks—get off to a fast start.

Attend all classes—your teacher needs you there in order to "teach you."

Work all assignments as they are assigned. Mathematics courses are usually building
courses, meaning each section builds on the concept that you have worked and understood
all of the material in the previous section(s).

Complete only one assignment at a time (in a block of time). Allow time for each
assignment to "soak in" before attempting the next assignment.

Ask questions. Prepare a list of questions to ask in class, if the teacher allows, or in the
teacher’s office or in a math help center.

Get help outside of class. You many want to go your teacher’s office or you may prefer to
go to a help center.
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Form study groups with 3 or 4 other students from your class.
III. Prior to a test:
•Plan ahead—set aside plenty of study time, beginning several
days before the test date.
•Go through the material presented/homework assignments.
Select 2 or 3 problems of each type presented and write yourself
a "practice test."
•For any areas that you feel are weak, select additional practice
exercises and/or get help on these areas.
•1-2 hours prior to taking the test, work a few problems—get
your mind thinking mathematics.
IV. After a test:
•Go over your test.
•Study any questions which you missed—learn how
to answer (work) these questions immediately.
•Keep your old tests in a safe place, as you will want
to study them again prior to the final exam.
V. How to approach homework assignments:
Read back through your class notes, including reworking all
examples, before you attempt your homework assignments and
read through the reading material in the text, working the
examples in the text.
As you are working through your homework, check your
answers. Remember that you can check your answers to the odd
problems from the back of the text.
If you miss a problem, try reworking it rather than trying to find
your mistake. Then compare your work. If you still miss it, look
for a similar example in the reading material in that section.
Using your own pencil and paper, work through that example and
compare to the example in the text.
Ask for HELP!
Properties of Real Numbers
Rules of Signs
a(-b) = -(ab) = (-a)b (-a)(-b) = ab - ( -a) = a
a a a
 
b b b
a a

b b
Properties of Real Numbers
Cancellation Properties
ac bc
ac
a

bc
b
implies a  b
if
if
c 0
b  0, c  0
Zero – Product Property
If ab = 0, then a = 0 or b = 0 or, both.
Properties of Real Numbers
Arithmetic of Quotients
a c
ad  bc


b d
bd
a c ac
 
b d bd
a
b  a  d  ad
c b c bc
d
if b  0 , d  0
if b  0, d  0
if b  0, c  0, d  0
Arithmetic of Quotients: Example 1
3 2 3  2 3  2
      
5 3 5  3 5 3
9  10
3 3 2 5
  
  
15 15
5 3 3 5
9  ( 10)
1


15
15
Arithmetic of Quotients: Example 2
12
5  12  20  12  20
3
5 3
5 3
20
4  3 5 4

 44
5 3
 16
The Real Number Line
The negative real numbers are the coordinates
of points to the left of the origin 0.
The real number zero is the coordinate of the
origin O.
The positive real numbers are the coordinates of
points to the right of the origin O.
Example of Domain
Find the domain of the variable z in
13
the expression
z+ 3
Domain:
z z 
3
The result is read “The set of all real numbers z
such that z is not equal to –3”
2
2
a + b = c
2
Example: The Pythagorean Theorem
Show that a triangle whose sides are of lengths 6,
8, and 10 is a right triangle.
We square the length of the sides:
2
6 = 36
2
8 = 64
2
10 = 100
Notice that the sum of the first two squares (36
and 64) equals the third square (100). Hence
the triangle is a right triangle, since it satisfies
the Pythagorean Theorem.
Geometry Formulas
For a rectangle of length L and width W:
Area = lw Perimeter = 2l + 2w
For a triangle with base b and altitude (height) h:
1
Area = bh
2
For a circle of radius r (diameter d = 2r)
Area = p r 2
Circumference = 2p r = p d
Geometry Formulas
For a rectangular box of length L, width W, and
height H:
V o lu m e = lw h
For a sphere of radius r:
4 3
Volume = p r
3
Surface
Area = 4p r 2
For a right circular cylinder of height h and radius r:
2
Volume = p r h
Exponents: Basic Definitions
If a is a real number and n is a positive integer,
a  a
 a

a



n
n facto rs
a 1
0
a
n
1
 n
a
if a  0
if a  0
Examples:
4  444
3
6 1
0
4
3
1
 3
4
Laws of Exponents
a a a
m n
mn
a 
m n
a
mn
ab  a b
m
a
1
mn
 a  nm if a  0
n
a
a
n
n
 a  a if b  0
 b bn
 a 
 b
n
b

 
 a
n
if a  0, b  0
n
n n
Example:
3 2
x y
Write 1 4 so that all exponents are positive.
x y
3 2
3
2
x y
x y
 1  4
1 4
x y
x
y
x
3( 1) 2  4
y
4 6
x y
4
x
 6
y
Example:
Simplify the expression. Express the answer so
only positive exponents occur.
 3x y 
 3 
 x y 
2
3
2
4
2
 3x
x   y 
1  2
23
3 2
y

4 1 2
  3x y
1
2
2
3 x y
2
6

3 2
x
 6
9y
Using your calculator
For Scientific Calculators:
Evaluate: (3.4) 4
Keystrokes: 3.4
xy
3.4
=
133.6336
For Graphing Calculators:
Evaluate: (3.4) 4
Keystrokes: 3.4
^
4
=
133.6336
Monomial
3x
4
2x
9
Coefficient
3
Degree
4
2
1
-9
0
Coefficients: 2, 0, -3, 1, -5
Degree: 4
2 x
3
 
 3x   x
 x  8 x  1  3x  5x  2
2

 2x
3
3
3
2

 8 x  5x   1  2
 5x  x  3x  1
3
2 x
3
2
 
 x  8 x  1  3x  5x  2
2
3

 2 x  x  8 x  1  3x  5x  2
3

2
 2 x  3x
3
3
3
 x
2
 8 x  5 x   1  2 
  x 3  x 2  13 x  3
3x  2 x  4 x  3
2
 3x x  3x  4x  3x  3 2 x  2  4x  2 3
2
2
 3 x  12 x  9 x  2 x  8 x  6
3
2
2
 3 x  14 x  17 x  6
3
2
The process of expressing a polynomial
as a product of other polynomials is
called factoring.
Example
(
2
Multiply: 3x x - 2 x - 4
)
2
= 3 x( x ) - 3 x(2 x) - 3 x(4)
= 3 x 3 - 6 x 2 - 12 x
Factoring is the same process in reverse
Factor: 3 x 3 - 6 x 2 - 12 x
Notice that each term in this trinomial has
a greatest common factor of 3x.
3 x 3 - 6 x 2 - 12 x
= 3x( x 2 ) - 3x(2 x) - 3x(4)
= 3x (x 2 - 2 x - 4)
Special Formulas
When you factor a polynomial, first check
whether you can use one of the special
formulas shown in the previous section.
Difference of Two Squares: x 2 - a 2 = ( x - a )( x + a )
Perfect Squares:
2
2
2
x + 2ax + a = (x + a )
2
x 2 - 2ax + a 2 = (x - a )
Sum of Two Cubes: x3 + a3 = (x + a)(x 2 - ax + a 2 )
3
3
2
2
x
a
=
x
a
x
+
ax
+
a
(
)
(
)
Difference of Two Cubes:
Example:
Factor Completely: 9 x 2 - 64
9 x  64   3 x   8 2
2
2
  3 x  8 3 x  8
Factor Completely: x 4  1

 x


 1  x  1  x  1 
2
2

x

1
x
1
x 1
4
2
Factor the trinomial: x  11x  18
2
Look for factors of 18 whose sum is 11.
9  2  18
9  2  11
x  11x  18   x  2 x  9
2
Factor the trinomial: x  3x  10
2
Factors of -10
10, -1
Sum
9
5, -2
3
-5, 2
-3
-10, 1
-9
x  3x  10   x  5 x  2
2
Factor completely: 3x  7 x  6
2
3x  7 x  6   3x
2
 3 x
 3 x

2
3x  7 x  6  
3
x


 3x
 x

1 x 6
6 x 1
2 x 3
3 x
2
 3x + 1 x  6
 3x  1 x  6

 3x  6 x  1

 3x  6 x  1
2
3x  7 x  6  
3
x

2
x

3




 3x  2 x  3

 3x  3 x  2
 3x  3 x  2
Factor By Grouping:
2 x  10x  3x  15
3
2


 2 x  10 x  3x  15
3
2
 2 x  x  5  3 x  5
2


  x  5 2 x  3
2
In general, if a is a nonnegative real number,
the nonnegative number b such that b2 = a is
the principal square root of a and is denoted
by b =
a.
a a
2
Absolute Value is needed here, since the
principal square root produces a positive value.
Example: (- 4)2 =
16
= 4 = - 4
Product Property of Square Roots
a b  ab
Example:
18  9  2  9 2  3 2
Example:
50x 
3

25x 2 x
2
25 x
2
 5 x 2x
2x
Rationalize the denominator in each expression
3
3
2
3 2



2
2 2
2
a 3
a 3
a  3
a 3 a 3




a 3 a 3
a9
2
The principal nth root of a real number a,
symbolized by n a is defined as follows:
n
a  b means a  b
n
where a > 0 and b > 0 if n
is even and a, b are any
real numbers if n is odd
n
a  a,
if n is odd
n
a  a,
if n is even
n
n
Examples:
3
81  9
because 9 2  81
 27  3 because - 3  27
3
Examples:
5
12  12
5
  12    12
6
5 5
6
 5   5  5
8
z  z
5
5
6
6
8
Properties of Radicals
n
ab  a b
n
n
n
n
a
a
n
b
b
n
a   a
mn
m
a
n
mn
a
m
Simplify:
4
32 x 
5
4
16x 2 x
 16 x
4
4
44
2x
 2 x 4 2x
Simplify:
8 x  3 x 50 x
3
 2 4  x  x  3x 25 2 x
2
= 2 x 2 x - 15 x 2 x = - 13x 2 x