MAT 108 Intermediate Algebra
Unit 4 Part 1
Summary of Factoring
In algebra, the instruction is usually not written as find the prime factorization of an
expression; we ask to factor completely. Its the same thing.
Greatest Common Factor  GCF
The first st
Mat 108 Intermediate Algebra
Unit 4 Part 5
Equations that Lead to Quadratic Equations
There are numerous examples of equations in mathematics that lead to quadratic
equations; we will study two of these. The first type is rational equations.
RATIONAL EQUA
Mat 108 Intermediate Algebra
Unit 4 Part 6
Higher Degree Equations in One Variable
The last topic for this unit is that of higher degree equations in one variable.
By our definition, these are equations where the exponent with the variable is greater
than
Mat  108 Intermediate Algebra
Unit 2 Part 3
LINEAR EQUATIONS IN TWO VARIABLES
EXAMPLES:
2 x 3 y 4
2 x 3 y 6 0
2 x 7 0
RECOGNITION:
(1) The highest exponent with either variable is one (implied)
(2) There is an equal symbol in the original problem
(3) The
Mat 108 Intermediate Algebra
Unit 4  Part 2
Quadratic Equations in One Variable
Although solving quadratic equations in one variable was discussed in the last unit (in
the radical equations), this unit is a more detailed study of this important topic. Le
MAT 108 Intermediate Algebra
Unit 4 Part 1
Factoring
Factoring is the skill of finding numbers or expressions that, when multiplied, give you
the original number or expression. For example, in arithmetic if we were asked to factor
12, we could say 4 3 . I
Mat 108 Intermediate Algebra
Unit 2 Part 5
Graphing Linear Inequalities
The graph of a linear equation in two variables is the graph of all ordered pairs that
satisfy the equation and the resulting graph is a line.
Now we are going to graph linear inequal
Mat 108 Intermediate Algebra
Unit 3 Part 7
Imaginary and Complex Numbers
If you remember, we encountered the square root of a negative number in the first section
of this unit specifically the  9 . At that time, we said that this was not a real
number it
Mat 108 Intermediate Algebra
Unit3 Part 6
Radical Equations in One Variable
This topic involves solving algebraic sentences and not manipulating algebraic
expressions; therefore, we will find solution sets rather than simplified expressions.
Remember that
Mat 108 Intermediate Algebra
Unit 4 Part 4
Quadratic Equations and the Quadratic Formula
Apparently, a mathematician (maybe Bombelli in the 16th century) tired of solving
quadratic equations by completing the square and decided to shortcut this repetitive
Mat 108 Intermediate Algebra
Unit 4 Part 3
The Square Root Method and Completing the Square
(Repeating some of the Notes from Part 2)
A long time ago, mathematicians solved quadratic equations of this last type (the
ax 2 +c =0 where there is no bx or no l
Mat 108 Intermediate Algebra
Review for Exam 2
Part 1
Cartesian Coordinate System
Be familiar with Cartesian coordinate system. Know how to plot points on
coordinate system.
Ordered Pairs
Each point, having an x and a y coordinate is called an ordered pai
Additional Review Problems for Unit 2 Exam
Find the equations of the following lines.
(1) Find the General Form of the equation of the line passing through
(  9,1) and (  1,  5 )
Answer:
3 x +4 y +23 =0
(2) Find the Standard Form of the equation of the
Comments on Solutions for Systems of Equations
Look at the chart below.
If a system of equations has one solution, as shown in the left diagram, the system is
considered to be consistent with independent equations,
If the system has no solution (as in the
Additional Problems for Unit 1 Review
1. Write a linear equation in one variable that has
(a)
A solution set of cfw_ or
(b)
A solution set of (or all real numbers)
(c)
A solution set with one value.
2. Write a linear absolute value equation in one varia
MAT 108 Intermediate Algebra
Review for Exam 4
Summary of Factoring
In algebra, the instruction is usually not written as find the prime factorization of an
expression; we ask to factor completely. Its the same thing.
Greatest Common Factor  GCF
The firs
Mat 108 Intermediate Algebra
Unit 4
Summary of Quadratic Equations in One Variable
(1) What does a quadratic equation in one variable look like?
An equation where the variable appears with an exponent of 2; there is an equal symbol;
there is only one vari
Mat 108 Intermediate Algebra
Unit 3 Part 5
Combining Radical Expressions with Division
As you will see with the notes for this assignment, we do not really combine radical
expressions with division so much as we simplify (rewrite) them.
Dealing with radic
Mat 108 Intermediate Algebra
Unit 3 Part 4
Combining Radical Expressions with Addition/Subtraction
There are numerous similarities between radicals (some of which are irrational) and
rational numbers (fractions) especially when it comes to the operations
Mat 108 Intermediate Algebra
UNIT 1  PART 3
Absolute Value Equations
Using a number line, we defined the absolute value of a number as its distance from
0 on the number line.
+5 five units from 0
units
() 5  five units from 0  the other direction, but
Mat 108 Intermediate Algebra
Unit 1 Part 2
Review Linear equations with one variable
We have been looking at linear equations in 1 variable, such as
1
x  1 =3x +9
2
Recognition: What do linear equations in one variable look like?
1. The variable appears
Mat 108 Intermediate Algebra
Review for Unit I Test
Algebraic Expressions (Unit I Part 1)
Be prepared to evaluate algebraic expressions.
Know the Order of Mathematical Operations shown below
Linear Equations in One Variable (Unit I Part 1)
Rules and strat
Homework Answers Unit 3 Part 1
I Pages 486487
Real Positive Numbers
(2)
20
(4)
3/5
(8)
3
(10)
x8
(12)
8y10
(14)
3.317
(18)
17.731
(22)
(28)
4x2
(32)
Not a real number
(34)
1
(38)
2x3
(40)
3x
(42)
16x4
(50)
x
Any Real Number
(47)
2x
(48)
2x
(52)

Homework Answers Unit 4 Part 1
Page 383
(10)
2(3x +2)(2 x  5)
(14)
6ab(4b  1)
(16)
9( x +3)( x  3)
(18)
(5 x +3)( x  1)
(20)
6( x  2)( x +1)
(22)
(5 x +4) 2
(24)
(3x  4 y )(9 x 2 +12 xy +16 y 2 (26)
(30)
(4r +5) 2
(32)
(5 x +2)(4 x +3)
(34)
20( x 
CRITIQUES AND
CRITICAL READING
Chapter 2
Critiques & Critical Reading
Require knowledge of the subject, imagination,
and your willingness to be involved in what
you read
Begin by asking yourself:
To what extent does the author succeed in his or
her purpos
Exam IV Practice Problems
Solve all equations algebraically over the complex. Leave all irrational and complex
solutions in simplified form. Write all complex solutions in a +bi form.
1.
15 x 2  23 x  5 =6 x 2  23x +11
4 4
Answer: ,
3 3
2.
6 x 2 =11x
Mat 108 Intermediate Algebra
Unit 1 Part 1
Evaluating Algebraic Expressions
Origins of Algebra
Algebra is a very old and powerful form of mathematics concerning the study of
structure, relation, and quantity of things, items
The word algebra is believed t