Who needs this manual?
What do you need to know to read this manual?
What will you learn?
Why should you trust me?
Identifying warnings and icons
Materials
1
1
1
2
2
3
Key Terms
Factoring Integers
The FOIL Method
5
6
7
x 2 + bx + c
Examples 12
ax 2 + bx + c
Example
18
Answer Key
21

If you are reading this manual, you are probably in a high school algebra class and may be thinking to yourself, “This isn’t fun, I have no idea what’s going on!”
If you’ve ever felt like this, or if you feel like math class is always moving at the speed of light and you can’t catch up, then this manual is for you!
This manual is designed for high school math students who want to improve their algebra skills by learning a quick and easy way to factor polynomials. “WHOA, slow down,” you might be saying. Don’t worry; I’ll explain everything as we go along. If you are an algebra student and you feel like you’re falling behind, then this manual is for you!
To benefit from these instructions on factoring, you must be in Algebra II or its equivalent. You should be learning about polynomials and their properties, such as how to multiply them together and how to break them apart. An important note is that this manual is not a substitute for class lectures or textbook instruction; it is simply a tool to make your life easier!
The steps I have outlined in this manual will teach you how to factor a quadratic equation using a technique I call the 4-Square Method. This method will help you remember how to factor which will come in handy for the nearly every math class you’ll have to take from now on! As scary as it sounds, it’s true…but after these simple steps you will have the tools necessary to be a successful mathematician in the future.
This manual is best read from start to finish the first time to ensure all the information has been covered. After that, it can be used as a reference for homework and extra practice.
1

I know you may be skeptical that this manual can really help your algebra skills.
A lot of math books are very confusing and hard to learn from. But the method I will teach you has personally helped in my math career, and because of it I’m now about to graduate from college with a math degree. I know what you’re thinking, and I never thought I would end up where I did either. Math wasn’t always an easy ride for me so I’m dedicated to making it easier and more fun for everyone!
Who knows, maybe you’ll change your mind about math one day…and if you do you’ll be ready to go far!
In the steps of this manual, there will often be warnings or helpful hints to make your experience with factoring easier. These icons are as follows:
This icon represents a “quick note.” You will see this icon after some of the steps in the directions section. If you are unsure what a step is telling you to do, or you need more clarification, look to this icon for help.
This warning symbol is included to remind you that you may get frustrated while reading this manual! Like any new concept, factoring can often be difficult to grasp at first. When you come to this icon, make sure to take a deep breath and be CONFIDENT that you will succeed…just keep going!
(Source: http://www.blogcatalog.com/images/usr/70372.gif)
This symbol means that it’s your turn to try! When there is an example for you to do on your scratch paper, you will see this icon and the answer to the example can be found at the bottom of the page.
2

You will only need a few materials before you read this manual. Make sure you are ready and prepared when you begin.
#2 Pencil(s)  Always use a pencil so you are able to correct mistakes neatly
Giant eraser  This may come in handy!

Blank scratch paper 
For practicing while following the steps in the manual
Good attitude  This one seems obvious, but don’t forget!
After you’ve gathered these materials, you can begin.
This is a perfect time to take a deep breath! Ready to start??
3

Even if we learn an important math concept, we often forget it soon after because we never use it or apply it to our lives. If our long-term memories don’t make any connections, the information could be lost until it is reviewed again. Some of the following review exercises are simple things you may have forgotten, but you’ll need these skills to learn proper methods of factoring.
If these review concepts are things you already know and are comfortable with, you can skip ahead to the next section.
4

The following are important terms along with their definitions. These terms will all be used throughout the manual, so make sure you understand them and know what they mean!
Polynomial  an algebraic expression consisting of one or more summed terms, each term consisting of a constant multiplier and one or more variables raised to integral powers. For example, x 2 - 5x + 6 and 2p 3 q + y are polynomials. Also called a multinomial.
Quadratic Equation  A specific type of polynomial… an expression consisting of the sum of two or more terms each of which is the product of a constant and a variable raised to the second power: ax 2 + bx + c is a quadratic, where a, b, and c are constants and x is a variable.
Factoring  to express a mathematical quantity as a product of two or more quantities of like kind, as 30 = 2·3·5, or x 2 − y 2 = (x + y) (x − y).
Another example: 6 and 3 are factors of 18
Cross Multiplying  used when multiplying fractions…the top of one fraction is multiplied by the bottom of the other. The results can be added together in some situations.
5

As you just learned in the Key Terms section, factoring is the process of breaking something apart into its different components. The purpose of this manual is to be able to break apart a quadratic into its factors…but for now, we will review how to factor only numbers (no algebra yet!).
To factor an integer, simply list all of the numbers can be divided evenly into the number. Do not list any number more than once.
Try to think of pairs of numbers that multiply together to equal your integer.
For example, let’s list the factors of 4. We can make 4 by multiplying 1 and
4 OR 2 and 2. So 4’s factors are 1, 2 and 4.
YOUR TURN! List the factors of 30.
(See answer below)
When factoring polynomials is discussed later in the manual, you will need to be able to factor numbers to find the algebraic factors of a quadratic. If you feel like you need more practice, choose 5 more numbers and list their factors.
ANSWER: 1, 2, 3, 5, 6, 10, 15, 30
6

In this manual you will be learning how to split equations apart; but do you know how to multiply the factors back together to check your answer?
FOIL is a way to remember how to multiply 2 factors together. It stands for
F irst
O uter
I nner
L ast
This means that you multiply the first terms together, then the outer and inner terms, and finally the last terms. Add like terms together for the answer.
Below is an example of how the FOIL method works. The factors and their products are color-coded so you can keep the terms straight.
EXAMPLE
Use the FOIL method:
(x+2) (x-4)
1. Multiply the first terms together. x*x=x²
2 . Multiply the outer terms, followed by the inner terms. Then add them together because they will both have an x in them (like terms). 2x+(-4x)=-2x
3. Multiply the last terms together. 2*(-4)=-8
By using the FOIL method, we find that (x+2) (x-4) = x²-2x-8.
7

YOUR TURN! Use the FOIL method to find the quadratic equation which is made up by (x+5) and (x+3).
WORK SPACE
Keep practicing FOIL as much as you can! This will be a great way to selfcheck your work when you complete this manual.
This is the end of the review section. If you feel unsure about any of the concepts mentioned here, look to a teacher or textbook for more information before moving on!
ANSWER: X²+8X+15
8

The rest of this manual will teach you how to factor a quadratic equation using the 4-Square Method. This method is easy to remember and easy to set up.
Remember the days of playing tic-tac-toe? The grid we will make is similar to the tic-tac-toe board, but with only 4 squares. This grid will help us find the factors for any quadratic equation.
It will look like this:
Don’t worry if you are a little confused right now! This is just to show you the process we will use.
9

x 2
bx
c
Now that you have had a little taste of my version of factoring, let’s begin! The first kind of quadratic we will focus on is the kind with no coefficient (leading number) in front of the x² term.
The 4-Square Method you were introduced to on the previous page will be handy now. First I will show the steps of this method, then some examples, and then it will be your turn to try!
Put an x in each of the left-hand boxes.
This will be done for you throughout most of this manual so you’ll know how to get started in the future.
Think of 2 numbers that will multiply to c in the original equation. Write these numbers in the right-hand boxes of the diagram.
If you are having trouble thinking of which numbers to put in the boxes, review methods of factoring integers
(page 6).
10

Write the answer as two factors. It should look like this: (x+f1)(x+f2)
The factors will be read straight across.
This means the top line is the first factor, and the bottom line is the second factor. Referring to the picture,
we can see why the factors are written
as (x+f1) (x+f2).
For these kind of quadratic equations, order doesn’t matter! f1 and f2 are interchangeable.
Now check your answer. If you multiply f1 and f2 together, do you get c? You can also use the
FOIL method to see if you get the original equation.
If your answer is incorrect, go back to step 2 and try again!
If you are feeling frustrated, it’s OK! Remember that it takes practice to master new concepts in math.
11

Let’s try a few examples using a real quadratic equation so you can try using the
4-Square Method yourself!
EXAMPLE #1
Factor x²+4x+3.
1.
Draw a grid to write your factors in. (Note the x’s in the left-hand boxes.)
2.
Find two numbers that multiply to 3 and add up to 4.
 List the factors of 3. The only ones are 1 and 3, so these numbers go in the empty boxes.
3. Then write the factors from right to left.
 (x+3) and (x+1)
4. FOIL to check your answer.
 (x+3) (x+1) = x²+3x+x+3 = x²+4x+3
12

EXAMPLE #2
Factor x²-6x+9.
1.
Draw a grid to write your factors in.
2.
Find two numbers that multiply to 9 and add up to -6.
 List the factors of 9. They are 1,3, and 9. Since we want -6, we can write -3 in both right-hand boxes.
3.
Then write the factors from right to left.
 (x-3) and (x-3)
4.
FOIL to check your answer.
 (x-3) (x-3) = x²-3x-3x+9 = x²-6x+9
13

WORK SPACE
YOUR TURN! Find the factors of x²+2x-15.
ANSWER: (x+5) (x-3)
14

x 2
bx
c
This section of the manual will teach you how to factor a quadratic equation when there is a leading coefficient (the number a in the equation above). The reason we use the 4-Square Method is because we will be cross multiplying on the grid to find our factors.
Factoring this type of quadratic equation is almost the same as before…even though it seems a lot harder!
Find two numbers that multiply to the 1 st term of the equation ax 2 + bx + c. (The ax 2 part.)
Write these in the red boxes.
If you have trouble, list the factors of the number. Then don’t forget to add an x on the end!
Now find two numbers that multiply to the c part of the equation (the constant term at the end). Place these in the blue boxes.
15

Now cross multiply the red and the blue boxes. Add the results together.
Forget how to cross multiply??
Start with the red box in the upper left corner. Multiply by it by the blue box in the lower right corner. Repeat with the other two boxes. Add the
results together.
When you added in step 3, was your answer the same number as the 2 nd term in the original equation (the bx part)?
If not… it’s OK! Just go back to step 1 and try different combinations of numbers.
This process of guessing and checking may seem tedious, but it is the best way to learn. Eventually it will become easier and take less trial and error to reach the right
answer.
Write the answer as two factors.
Again, the factors will be read straight across. This means the top line is the first factor, and the bottom line is the second factor.
For these kind of quadratic equations, order DOES matter! f1 and f2 must be in the right place for the factors to FOIL correctly.
16

Now check your answer. You can use the FOIL method to see if you get the original equation.
If your answer is incorrect, go back to step 1 and try again!
Now we will look at a specific example so you can see how this works. Then there are a few exercises for you to try at the end of the section.
17

EXAMPLE
Factor 2x²+x-21.
1.
Find two numbers that multiply to 2x².
 Since 2 has only two factors, 1 and 2, we will write x and 2x in the red boxes.
(If we multiply these together, we will get the first term of the equation, 2x².)
2.
Then find two numbers that multiply to -21. (Note the negative!)
 The factors of 21 are 1, 3,7,21. Let’s try 3 and 7. Remember, one of these has to be negative to multiply back to -21.
3.
Now we will cross multiply and add to see if we chose the right numbers.
 Our answer should add up to the second term of the equation, x.
( x )

( 7 )

7 x
( 2 x )

(

3 )
 
6 x
7 x

(

6 x )
 x
Since we got the right answer in step 3, we have found our factors, (x-3) & (2x+7).
 FOIL to make sure the answer is right:
( x

3 )( 2 x

7 )

2 x
2

6 x

7 x

21

2 x
2
 x

21
18

Congratulations, you have made it through the manual! Whether you are 100% confident in factoring yet or just getting comfortable, the following tips are for you. They are just a couple of common questions that are asked about factoring that can benefit anyone learning to factor.
Q:
What if a quadratic will not factor, no matter how many combinations of numbers I try?
A:
Remember that factoring is a process of trial and error. Sometimes you will have try a lot of different combinations! However, there are cases in which a quadratic equation cannot be factored. It is unlikely you will deal with any quadratics that do not factor using the 4-Square Method until other methods of finding roots are taught.
Q:
What if a quadratic equation has no middle term? Example: x²-64
A:
This is a special case of the quadratic called a difference of squares.
Luckily, there is any easy trick to factoring this kind of polynomial! If a quadratic looks like this (note the negative between terms):
Its factors look like this:
Using the example from above with this trick, we see the factors of x²-64 are (x+8)(x-8).
19

Factor these quadratic equations using the 4-Square Method. Copy the problems on another sheet of paper so you have room to work.
If you need blank grids, see page 22!
(1) x²+4x+3
(2) x² -4x-12
(3) x²+x+20
(4) x²+7x+6
(5) x² -5x+6
(6) x²+13x+40
(7) x² -4
(8) x² -x-2
(9) x²+2x+1
(10) x² -10x+21
(11) 8 x²+9x+3
(12) 3 x² -12x+12
(13) 2 x²+6x -8
(14) 2 x²+5x -3
(15) 3 x²+17x+10
(16) 5 x²+4x -9
(17) 12 x²+14x+4
(18) 2 x²+3x -14
(19) 6 x² -x-1
(20) 16 x² -1
20

1.
(x+1)(x+3)
2.
(x+2)(x-6)
3.
(x-4)(x+5)
4.
(x+1)(x+6)
5.
(x-3)(x-2)
6.
(x+5)(x+8)
7.
(x+2)(x-2)
8.
(x-2)(x+1)
9.
(x+1)(x+1)
10.
(x-7)(x-3)
11.
(2x+3)(4x+1)
12.
(x-2)(3x-6)
13.
(x+4)(2x-2)
14.
(3x+2)(x+5)
15.
(3x+2)(x+5)
16.
(x-1)(5x+9)
17.
(4x+2)(3x+2)
18.
(x-2)(2x+7)
19.
(3x+1)(2x-1)
20. (4x+1)(4x-1)
21

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