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Math 60 Chapter 5.notebook
October 20, 2014
MathB60 - Elementary Algebra
Fall 2014
Ms. Rush
LA 112 office
Chapter 5 - Course Notes
Oct 20-3:36 PM
1
Math 60 Chapter 5.notebook
October 20, 2014
MathB60
Chapter 5: Polynomials and Factoring
5.1
Introduction to Factoring
5.2
Factoring Trinomials of the Type x2 + bx + c
5.3
Factoring Trinomials of the Type ax2 + bx + c
5.4
Factoring Perfect-Square Trinomials and Differences of Squares
5.5
Factoring Sums or Differences of Cubes
5.6
Factoring: A General Strategy
5.7
Solving Polynomial Equations by Factoring
5.8
Solving Applications
May 20-11:03 PM
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Math 60 Chapter 5.notebook
October 20, 2014
Factoring
To factor a polynomial is to find an equivalent
expression that is a product. An equivalent
expression of this type is called a factorization of
the polynomial.
Recall:
Last chapter we multiplied polynomials together using:
1. Distributive Property
2. FOIL
3. Patterns
NOW: Factoring will reverse this process. Factoring is reverse distributive property.
Challenge: During the time we spend in this chapter, try factoring everyday!
Perfection comes with a lot of practice.
May 20-11:13 PM
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Math 60 Chapter 5.notebook
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5.1 Greatest Common Factor (GCF)
What are the factors of 24?
What are the factors of
What are some factorizations of 24?
?
What are some factorizations
of
?
May 20-11:25 PM
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Math 60 Chapter 5.notebook
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5.1 Greatest Common Factor (GCF)
What are the factors of 24?
What is the GCF of 24 & 32?
What are the factors of 32?
At this point in the course, we have used the Distributive Law for
multiplication. Use 'reverse distributive property' to factor out a GCF.
May 20-11:25 PM
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Math 60 Chapter 5.notebook
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Ex: Factor out the GCF by using 'reverse distributive property'
1) 9a - 21
4)
2)
5)
3)
May 20-11:48 PM
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Math 60 Chapter 5.notebook
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Ex: Factor out a negative as part of the GCF.
We typically will factor out a negative if the leading term is negative.
1).
4).
2)
5)
3)
6)
May 20-11:48 PM
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Math 60 Chapter 5.notebook
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Binomials as GCF
or
Consider
GCF =
GCF =
Ex: Factor
1) x(3x - 2) - 8(3x - 2)
3) 2x(5x + 7) + 3(5x + 7)
2) x(7x + 1) + 2(7x + 1)
4) 6x(5 - x) - (5 - x)
May 20-11:55 PM
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Math 60 Chapter 5.notebook
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Factor by Grouping
Ex: Factor
1)
Consider
2)
What is the GCF?
In this form, there is no common factor to factor out. There is
another method that could possible work, called factoring by
grouping.
May 21-10:10 AM
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Math 60 Chapter 5.notebook
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Ex: Factor by Grouping.
1)
2)
3)
4)
5)
6)
May 21-12:01 PM
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Math 60 Chapter 5.notebook
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5. 2 Trinomials of the form
Ex: Multiply
1) (x + 2)(x + 6)
4) (x - 2)(x - 6)
2) (x + 3)(x + 4)
5) (x - 3)(x - 4)
3) (x + 1)(x + 12)
6) (x - 1)(x - 12)
May 21-12:12 PM
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Math 60 Chapter 5.notebook
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L=c
The Pattern
I
O + I = bx
O
So to factor the form
product is c and sum is b.
, we need two numbers whose
May 21-1:11 PM
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Case where both b & c are positive.
Ex: Factor
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Case where c is negative & b is positive.
Ex: Factor
May 21-4:05 PM
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Ex: Factor Completely
May 21-4:11 PM
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5.3 Trinomials of the form
Finding a Pattern!
F
L
What part of the multiplication
does this term come from?
May 21-4:04 PM
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Trial & Error Method
"Try"
F=
(
L=
)(
)
"Check if O + I = bx
May 21-6:31 PM
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Example:
F=
&
L = -5
Possible Factorizations
(3x - 5)(x + 1)
(3x - 1)(x + 5)
(3x + 5)(x - 1)
(3x + 1)(x - 5)
May 21-6:35 PM
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Ex: Factor. Write out all possibilities.
May 21-7:19 PM
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Ex: Factor by guess and check method. Know how many
possibilities to try!
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Math 60 Chapter 5.notebook
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Ex: Factor by guess and check method. Know how many
possibilities to try!
May 21-7:19 PM
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A second method to factor form
: ac method (by grouping)
Step 1: Find two numbers whose
product = ac & sum = b.
Step 2: Re-write
but replacing bx term as a
sum using numbers in step 1.
Step3: Factor by grouping.
Ex: Factor
Step 1:
Step 2:
Step 3:
May 21-8:44 PM
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Ex: Factor Completely using the AC-method.
1)
3)
2)
4)
May 21-8:53 PM
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Ex: Factor Completely using the AC-method.
5)
6)
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5.4/5.5 Special Forms
Recall: Multiply Polynomials
Factored form is given when we use these 'backwards"
from the techniques of chapter 4.
May 21-8:58 PM
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Ex: Factor
May 22-10:41 AM
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Ex: Factor Completely
May 22-10:45 AM
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5.4/5.5 Special Forms
Ex: Factor by identifying 'a' and 'b' and using the special patterns.
May 21-8:58 PM
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5.4/5.5 Special Forms
Ex: Factor by identifying 'a' and 'b' and using the special patterns.
May 21-8:58 PM
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Sum or Difference of Cubes
Perfect Cubes
1
8
27
64
125
216
May 22-8:20 PM
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Ex: Multiply, using the distributive property.
1)
2)
Sum/Difference of Cubes
May 22-10:50 AM
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Pattern Method to Factoring Sum or Difference of Cubes
1
Second
First
4
2
product w/
sign change
3
May 22-8:18 PM
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Ex: Factor
May 22-8:32 PM
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Ex: Factor Completely
May 22-8:35 PM
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5.6 Factoring Strategy
Strategy to Factor Completely
Step 1: GCF
Step 2: Look at number of terms.
Two Terms
Three Terms
Method in section 5.2
Method in section 5.3
Four or More Terms
Try Factor By Grouping
Step 3: Check to see if any factors will factor further
May 22-8:38 PM
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Ex: Factor Completely.
May 22-8:50 PM
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Math 60 Chapter 5.notebook
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Ex: Factor Completely.
May 22-8:52 PM
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Ex: Factor Completely.
May 22-9:56 PM
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Ex: Factor Completely.
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5.7 Solving Quadratic Equations by Factoring
Definition: Standard Form of a Quadratic Equation
Remark: The strategy we applied to solve linear
equations will NOT work on this type of equation.
The Principle of Zero Products
Ex: Solve (x - 3)(x + 5) = 0
May 22-10:05 PM
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Math 60 Chapter 5.notebook
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Ex: Solve.
Note: It is difficult to see solutions
while equation is in this form!
Strategy to solve quadratic (or higher degree polynomial)
Step 1: Set one side to zero
(Write equation in standard form).
Step 2: Factor.
Step 3: Set each factor to zero & solve.
May 22-10:12 PM
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Ex: Solve.
May 22-10:18 PM
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Ex: Solve.
May 22-10:21 PM
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Ex: Solve.
May 22-10:23 PM
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5.8 Applications
Ex: Solve using Algebra
The product of two facing pages in a book is 506. Find the
page numbers.
May 22-10:27 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
The sum of the square of two consecutive even numbers is 340.
Find the numbers.
May 22-10:32 PM
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Math 60 Chapter 5.notebook
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There is a special theorem that relates the sides of a right triangle.
Hy
po
ten
us
e
Right
Angle
Legs
Pythagorean Theorem
c
a
b
May 22-10:35 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
A 13-ft ladder is leaning against a house. The distance from the bottom of the
ladder to the house is 7 ft less than the distance from the top of the ladder to the
ground. How far is the bottom of the ladder from the house?
May 22-10:32 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
The math club is designing a brochure. The design calls for a triangle to be
placed on the front. The triangle has a base that is 6 centimeters less than the
height. If the area of the triangle is 216 cm2. Find the height and base.
May 22-10:32 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
The Mitchell’s are designing a garden. The garden will be in the shape of a
rectangle and have an area of 270 square feet. The width of the garden is 3 feet
less than the length. Find the length and width.
May 22-10:45 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
A rectangle is formed from a square by adding 6m to one side and 3m to the
other side. The area of the rectangle is 238 m2. Find the dimensions of the
original square.
May 22-10:45 PM
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Math 60 Chapter 5.notebook
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Ex: Solve using Algebra
A rectangular lawn is 14m long and 10m wide. A strip of uniform width is
mowed around the outside of the lawn. How wide is the strip when the area of
the unmowed portion of the lawn is 12m^2?
May 22-10:45 PM
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