1
Note-Taking Guides
How to use these documents for success
Print all the pages for the module.
Open the first lesson on the computer.
Fill in the guide as you read.
Do the practice problems on notebook paper (usually).
Put the notes and practice problems in a notebook. You can use these anytime!
Review the notes before you go to sleep. Short term memory is converted to long term
memory ONLY while you sleep. Your brain starts at the end of your day and converts
things to long term memory in reverse order.
2
Module #7
7.01 GCF
Factoring
Fill in the factoring flowchart below:
Factoring out the GCF
There are three steps to finding the GCF and factoring an expression. They are:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
3
Example 1:
2x + 6
Steps
Step 1 – Identify GCF
Work
Step 2 – Factor out GCF
Step 3 – Write the answer
Example 2:
10x3 + 15x2
Steps
Step 1 – Identify GCF
Work
Step 2 – Factor out GCF
Step 3 – Write the answer
Example 3:
Steps
Step 1 – Identify GCF
Step 2 – Factor out GCF
Step 3 – Write the answer
8x3 - 4x2 – 10x
Work
4
Example 4:
-6x4y – 9x3y2 + 3x2y3
If the first term in the expression you are factoring is negative, be sure to: _________________
____________________________________________________________________________
Steps
Step 1 – Identify GCF
Work
Step 2 – Factor out GCF
Step 3 – Write the answer
Special Note
A prime number has only two factors. They are:
1. ______________________________________________________
2. ______________________________________________________
If an expression does not have a common factor, it is called a ________________________ .
5
7.02 Difference of Squares
A difference of squares is a polynomial that:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
Perfect Squares
Perfect squares are numbers that: _______________________________________
The exponents on variables of perfect squares are: __________________________
The formula for factoring the difference of squares is: _________________________________
Remember: if you have a GCF in a difference of squares problem you must factor that out first.
It’s always step 1 in our factoring process.
Example 1:
Factor 9x2 – 4
Steps
Step 1 – Identify the perfect squares
Work
Step 2 – Use formula to factor
Example 2:
Factor x2 – 36y4
Steps
Step 1 – Identify the perfect squares
Step 2 – Use formula to factor
Work
6
Example 3:
Factor 2x2y2 – 50
Steps
Step 1 – Identify the perfect squares
Work
Step 2 – Use formula to factor
If you have a GCF in a problem, you must __________________________________________
7
7.03 Factoring by Grouping
The four steps to factoring by grouping are:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
Example 1:
Factor 7x3 - 28x2 + 3x - 12
Steps
Step 1 – Divide the polynomial
Work
Step 2 – Factor out GCF (first
terms)
Step 3 – Factor out GCF (last
terms)
Step 4 – Factor the parentheses
Example 2:
Factor 6x3 + 3x2 - 2x - 1
Steps
Step 1 – Divide the polynomial
Step 2 – Factor out GCF (first
terms)
Step 3 – Factor out GCF (last
terms)
Step 4 – Factor the parentheses
Work
8
Example 3:
Factor 60ab – 20bx – 30ax + 10x2
Steps
Step 1 – Divide the polynomial
Work
Step 2 – Factor out GCF (first
terms)
Step 3 – Factor out GCF (last
terms)
Step 4 – Factor the parentheses
Parentheses don’t match!
The two possible reasons for the parenthesis not to match in this process are:
1. ______________________________________________________
2. ______________________________________________________
9
7.04 Factoring Trinomials – Part 1
Factoring Trinomials of the form x2 + bx + c
Factoring trinomials is like what process in reverse? __________________________________
Steps to Factoring Trinomials
There are four steps to factoring trinomials of the form x2 + bx + c:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
Example 1:
Factor x2 + 6x + 8
Steps
Step 1 – Identify all the factor pairs
for the last term
Step 2 – Select the pair of factors
that add up to the middle term
Step 3 – Determine the signs
Step 4 – Check your guess using
FOIL
Work
10
Example 2:
Factor x2 - 12x + 35
Steps
Step 1 – Identify all the factor pairs
for the last term
Work
Step 2 – Select the pair of factors
that add up to the middle term
Step 3 – Determine the signs
Step 4 – Check your guess using
FOIL
Example 3:
Factor x2 - 3x - 18
Steps
Step 1 – Identify all the factor pairs
for the last term
Step 2 – Select the pair of factors
that add up to the middle term
Step 3 – Determine the signs
Step 4 – Check your guess using
FOIL
Work
11
Example 4:
Factor x2 + 2xy + y2
Steps
Step 1 – Identify all the factor pairs
for the last term
Work
Step 2 – Select the pair of factors
that add up to the middle term
Step 3 – Determine the signs
Step 4 – Check your guess using
FOIL
Sign Rules for Factoring Trinomials
If the last term in the trinomial is positive: ___________________________________________
If the last term of the trinomial is negative: __________________________________________
If the middle term is positive: ____________________________________________________
If the middle term is negative: ___________________________________________________
If the last term in the polynomial is negative: ________________________________________
12
7.06 Factoring Trinomials – Part 2
Factoring Trinomials of the form ax2 + bx + c
What is a “leading coefficient?” _________________________________________________
Factoring by Guessing and Checking
There are four steps to factoring trinomials of the form ax2 + bx + c by guessing and checking:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
Example 1:
Factor 8x2 + x - 7
Steps
Step 1 – Determine signs
Step 2 – Determine factors of first
term
Step 3 – Determine factors of last
term
Step 4 – Check your factors using
FOIL
Work
13
Factoring by Grouping
There are six steps to factoring trinomials by grouping:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
5. ______________________________________________________
6. ______________________________________________________
Example 2:
Factor 2x2 - 7x + 6
Steps
Step 1 – Multiply first and last
coefficients
Step 2 – List all factor pairs
Step 3 – Determine factors that
multiply to give last coefficient but
add to give middle coefficient
Step 4 – Write original trinomial as
4-term polynomial
Step 5 – Factor by grouping
Step 6 – Check using FOIL
Work
14
Example 3:
Factor 5x2 + 8xy - 4y2
Steps
Step 1 – Multiply first and last
coefficients
Work
Step 2 – List all factor pairs
Step 3 – Determine factors that
multiply to give last coefficient but
add to give middle coefficient
Step 4 – Write original trinomial as
4-term polynomial
Step 5 – Factor by grouping
Step 6 – Check using FOIL
Example 4:
Factor 8x2 – 14x + 5
Steps
Step 1 – Multiply first and last
coefficients
Step 2 – List all factor pairs
Step 3 – Determine factors that
multiply to give last coefficient but
add to give middle coefficient
Step 4 – Write original trinomial as
4-term polynomial
Step 5 – Factor by grouping
Step 6 – Check using FOIL
Work
15
7.07 Factoring Completely
10x3 – 40x
Example 1:
Steps
Work
Step 1
Step 2
Step 3
y5 - 2y4 - 35y3
Example 2:
Steps
Step 1
Step 2
Step 3
Work
16
2x2 - 11x + 12
Example 3:
Steps
Work
Step 1
Step 2
Step 3
3m4 - 3
Example 4:
Steps
Work
Step 1
Step 2
Step 3
2x4 – 5x2 - 12
Example 5:
Steps
Step 1
Step 2
Step 3
Work
17
7.08 Solving Equations by Factoring
Review of Linear Equations
Linear equations look like: ____________________________________________________.
The highest power in a linear equation is: ________________________________________.
Linear equations have ____________________________ and ______________________ on
the variables.
The graph of a linear equation is: ______________________________________________.
Quadratic Equations
A quadratic equation looks like: _________________________________________________
The highest power in a quadratic equation is: ______________________________________
Quadratic equations have __________________________ and _______________________ as
the highest power of x.
The solutions to quadratic equations are the points where: ___________________________
The graph of a quadratic equation is: ______________________________________________
Solving Quadratic Equations
Quadratic equations can be written in two forms:
1. ______________________________________________________
2. ______________________________________________________
In the second, the y is replaced with: __________________________.
Quadratic equations can have _____, _____, or no solutions.
Most of the time, there are ____ solutions.
The solutions are the _______________________ of the parabola.
Solving Quadratic Equations by Factoring
There are three General Steps to solving quadratic equations by factoring.
Step 1.____________________________________________________________________
Step 2.____________________________________________________________________
Step 3.____________________________________________________________________
18
Explain the property for zero that explains why we can solve quadratic equations by factoring:
___________________________________________________________________________
Example 1:
3x2 = 12x
Steps
Step 1 – Set equation equal to zero
Work
Step 2 - Factor
Step 3 – Set factors equal to zero
Example 2:
x2 – 7x + 12 = 0
Steps
Step 1 – Set equation equal to zero
Work
Step 2 - Factor
Step 3 – Set factors equal to zero
Example 3:
6x2 – 9x = 60
Steps
Step 1 – Set equation equal to zero
Step 2 - Factor
Step 3 – Set factors equal to zero
Work
19
Example 4:
x2 + 6x + 9 = 60
Steps
Step 1 – Set equation equal to zero
Step 2 - Factor
Step 3 – Set factors equal to zero
Work
20
7.09 Quadratic Formula
The quadratic formula allows you to find the values of x even when the quadratic polynomial is
not factorable.
The Quadratic Formula
The quadratic formula looks like:
The equation must be set equal to: __________________________________
The equation should be placed in: ___________________________________
The values for a, b and c are pulled from the quadratic equation being solved.
1. a is _____________________________________________________
2. b is _____________________________________________________
3. c is _____________________________________________________
Follow the order of operations when simplifying the quadratic formula.
When do most students make mistakes with the quadratic formula? ____________________
__________________________________________________________________________
Example 1:
2x2 + 3x – 6 = 0
Steps
Step 1 – Identify a, b and c
Step 2 – Substitute a, b and c in the
formula
Step 3 – Write the two solutions for
x
Work
21
Example 2:
x2 – 4x - 3 = 0
Steps
Step 1 – Identify a, b and c
Work
Step 2 – Substitute a, b and c in the
formula
Step 3 – Write the two solutions for
x
Example 3:
2x2 + 5x = 3
Steps
Step 1 – Identify a, b and c
Work
Step 2 – Substitute a, b and c in the
formula
Step 3 – Write the two solutions for
x
Example 4:
x2 - 2x + 1 = 0
Steps
Step 1 – Identify a, b and c
Step 2 – Substitute a, b and c in the
formula
Step 3 – Write the two solutions for
x
Work
22
7.10 Graphing Quadratic Functions
Sketching Graphs of Quadratics by Hand
There are 4 steps to graphing a quadratic function by hand:
1. ______________________________________________________
2. ______________________________________________________
3. ______________________________________________________
4. ______________________________________________________
To calculate the x-coordinate of the vertex, use the formula: ___________________________.
Example 1
Sketch the graph of the quadratic function y = 5x2
Steps
Step 1 – Standard Form
Work
Step 2 – Find vertex
Step 3 – Table of Values
x
y
23
Step 4 – Plot and sketch
Example 2
Sketch the graph of the quadratic function y = x2 + 4x + 3
Steps
Step 1 – Standard Form
Work
Step 2 – Find vertex
Step 3 – Table of Values
x
y
24
Step 4 – Plot and sketch
Example 3
Sketch the graph of the quadratic function y = -4x2 + 8x - 1
Steps
Step 1 – Standard Form
Work
Step 2 – Find vertex
Step 3 – Table of Values
Step 4 – Plot and sketch
x
y
25
Graphing Quadratics using Technology
Go to the website www.geogebra.org and download the software.
Why do we use technology to graph quadratic equations? ______________________________
The solution to a quadratic function is/are: __________________________________________