6.3
Factoring Trinomials of the
2
Form ax + bx + c and
Perfect Square Trinomials
Factoring Trinomials
Returning to the FOIL method,
F
O
IL
(3x + 2)(x + 4) = 3x2 + 12x + 2x + 8
= 3x2 + 14x + 8
To factor ax2 + bx + c into (#1x + #2)(#3x + #4), note
that a
3.4
Slope and Rate of Change
Slope of a Line
Slope of a Line
rise y change y2 y1
m=
=
=
run x change x2 x1
x2 x1
Martin-Gay, Beginning and Intermediate Algebra, 4ed
2
Slope of a Line
Slope of a Line
Positive Slope:
m>0
y
x
Negative Slope:
m<0
y
Lines wit
12.4
Logarithmic Functions
Graph of a Exponential Function
If we graph an exponential function where the
base > 1, we get an increasing function, as
y
shown below.
x
Martin-Gay, Beginning and Intermediate Algebra, 4ed
2
Graph of a Logarithmic Function
We
12.3
Exponential Functions
Exponential Expressions
We have previously worked with exponential
expressions, where the exponent was a
rational number
The expression bx can actually be defined for
all real numbers, x, including irrational
numbers.
However,
12.2
Inverse Functions
One-to-One Functions
We have studied functions, which are defined to
require that each element of the domain (input
values) produce a unique element of the range
(output values).
1-1 functions also require that each element of
the
11.5
Quadratic Functions
and Their Graphs
Graphing Quadratic Functions
We first examined the graph of f (x) = x2 back in
Chapter 3. We looked at the graphs of general quadratic
functions of the form f (x) = ax2 + bx + c in Chapter 5.
We discovered that t
11.3
Solving Equations by
Using Quadratic Methods
Solving Quadratic Equations
Solving a Quadratic Equation
1)
2)
3)
4)
If the equation is in the form (ax + b)2 = c, use the
square root property and solve. If not, go to Step 2.
Write the equation in stand
11.2
Solving Quadratic
Equations by Using the
Quadratic Formula
The Quadratic Formula
Another technique for solving quadratic
equations is to use the quadratic formula.
The formula is derived from completing the
square of a general quadratic equation.
Ma
Chapter 11
Quadratic Equations
and Functions
11.1
Solving Quadratic
Equations by Completing
the Square
Square Root Property
We previously have used factoring to solve
quadratic equations.
This chapter will introduce additional
methods for solving quadrat
10.7
Complex Numbers
Imaginary Numbers
Previously, when we encountered square roots
of negative numbers in solving equations, we
would say no real solution or not a real
number.
Imaginary Unit
The imaginary unit i, is the number whose square is 1.
That i
10.6
Radical Equations and
Problem Solving
The Power Rule
Power Rule
If both sides of an equation are raised to the same
power, solutions of the new equation contain all the
solutions of the original equation, but might also
contain additional solutions.
12.5
Properties of
Logarithms
Properties of Logarithms
This section examines several properties of
logarithms that allow you to simplify
expressions.
Recall that a logarithm is an exponent, so
logarithmic properties are restatements of
exponential proper
12.6
Common Logarithms,
Natural Logarithms,
and Change of Base
Common and Natural Logs
There are two logarithmic bases that occur so
frequently in applications that they are given
special names.
Common logarithms are logarithms to base 10.
Natural logari
MONTGOMERY COLLEGE
GERMANTOWN CAMPUS
MA 103 Final Exam Review Sheet
(Revised Spring 2009)
This is a review sheet. The actual final examination will be different.
1. Which of the following graphs
i) Represents a function? Give a reason for your answer.
ii)
College Level Mathematics
The college level section of the placement test tests you on skills which would allow you
to register for college level classes in college. Results of this portion will enable us to place
students in Algebra, Intermediate Algebra
Elementary Algebra
The Elementary Algebra version of the placement test has 12 questions and tests you in the
following areas. Your results on this section of the test will determine your placement into
MATH 061 or MATH 067.
Operations with integers and r
MONTGOMERY COLLEGE
Business, Science, Math and Applied Technologies Division
Germantown Campus
MA 116 Elements of Statistics
Course Syllabus
I. INSTRUCTOR INFORMATION
Name:
Dr. Munther Alraban
Telephone Number:
240-567-1972
E-Mail Address:
Munther.alraban
MONTGOMERY COLLEGE
Business, Science, Math and Applied Technologies Division
Germantown Campus
MA 116 Elements of Statistics
Course Syllabus
I. INSTRUCTOR INFORMATION
Name:
Dr. Munther Alraban
Telephone Number:
240-567-1972
E-Mail Address:
Munther.alraban
MONTGOMERY COLLEGE
Business, Science, Math and Applied Technologies Division
Germantown Campus
MA 116 Elements of Statistics
Course Syllabus
I. INSTRUCTOR INFORMATION
Name:
Dr. Munther Alraban
Telephone Number:
240-567-1972
E-Mail Address:
Munther.alraban
MONTGOMERY COLLEGE
GERMANTOWN CAMPUS
MA 103 Final Exam Review Sheet
(Revised Spring 2011)
This is a review sheet. The actual final examination will be different.
1. Which of the following graphs
i) Represents a function? Give a reason for your answer.
ii)
MONTGOMERY COLLEGE
GERMANTOWN CAMPUS
MA 103 Final Exam Review Sheet
(Revised Spring 2011)
This is a review sheet. The actual final examination will be different.
1. Which of the following graphs
i) Represents a function? Give a reason for your answer.
ii)
MONTGOMERY COLLEGE
GERMANTOWN CAMPUS
MA 103 Final Exam Review Sheet
(Revised Spring 2011)
This is a review sheet. The actual final examination will be different.
1. Which of the following graphs
i) Represents a function? Give a reason for your answer.
ii)
MONTGOMERY COLLEGE
GERMANTOWN CAMPUS
MA 103 Midterm Exam Review Sheet
(FALL 2010)
This is a review sheet. The actual examination will be different.
1. Which of the following graphs
i) Represents a function? Give a reason for your answer.
ii) Represents a
10.5
Rationalizing Numerators
and Denominators of
Radical Expressions
Rationalizing the Denominator
Many times it is helpful to rewrite a radical quotient
with the radical confined to ONLY the numerator.
If we rewrite the expression so that there is no
r
10.4
Adding, Subtracting, and
Multiplying Radical
Expressions
Sums and Differences
Rules in the previous section allowed us to
split radicals that had a radicand which was a
product or a quotient.
We can NOT split sums or differences.
a+b a + b
a b a b
M
6.6
Solving Quadratic
Equations by Factoring
Zero Factor Theorem
Quadratic Equations
Can be written in the form ax2 + bx + c = 0.
a, b and c are real numbers and a 0.
This is referred to as standard form.
Zero Factor Theorem
If a and b are real numbers
6.5
Factoring Binomials
Difference of Two Squares
Another shortcut for factoring a trinomial is when we
want to factor the difference of two squares.
a2 b2 = (a + b)(a b)
A binomial is the difference of two squares if
1.both terms are squares and
2.the s
6.4
Factoring Trinomials of
2
the Form x + bx + c
by Grouping
Factoring by Grouping
To Factor Trinomials by Grouping
1)
Factor out a greatest common factor, if there is
one other than 1.
2)
For the resulting trinomial ax2 + bx + c, find two
numbers whose
6.2
Factoring Trinomials of
2
the Form x + bx + c
Factoring Trinomials
Recall by using the FOIL method that
F
O
I
L
(x + 2)(x + 4) = x2 + 4x + 2x + 8
= x2 + 6x + 8
To factor x2 + bx + c into (x + one #)(x + another #),
note that b is the sum of the two n
Chapter 6
Factoring
Polynomials
6.1
The Greatest Common
Factor and Factoring by
Grouping
Factors
Factors (either numbers or polynomials)
When an integer is written as a product of
integers, each of the integers in the product is a
factor of the original