# The Greenebox Factoring Method

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The Greenebox Factoring Method
In Algebra, factoring has always been a very complicated procedure to learn as well as to teach. The main problem with the current way we teach factoring is there are too many methods to learn and each is used for a different type of polynomial. As if that weren’t complicated enough, there are at least seven formulas that need to be memorized as well. The Greenebox Factoring Method can be used to factor polynomials with four terms, three terms and even two terms (the difference of two squares). The major advantage of this method is it gives the student ONE METHOD that can be used to factor almost everything. The only thing left to memorize are the cube formulas. Then after they feel confident with their factoring skills, the teacher can show them other shortcuts and formulas. ã Copyright 1999 Lynda Greene all rights reserved

Greenebox Factoring Method- Four Terms
L 4 terms: Example 1: FIRST STEP: Draw a box and insert the four terms in the correct positions. Use the RED (FOIL) letters as your guide. F O I L (include the sign “+, -” to the left of each term) Note: When a student actually works the problem, there is no need to include the labels (in red) or the labels on the original (FOIL), it is enough to be able to identify which term is first, outer, etc. Notice that the sign of each term is included with it inside the squares of the box.

+ay - 2b -2bx y + ASK YOURSELF:
Looking at terms two at a time, what does each pair have in common? - 2b This is called the GCF (Greatest Common Factor) Write the GCF next to each row or column O F -2bx Take the signs of the outer or inner terms: O: take ‘-’ from -2bx I: and ‘+’ from +ay I L +ay + y Recall that the GCF is the largest factor each of the terms has in common in a particular row or column. Example: the two terms in the first row each have an ‘a’ in common The sign is always taken from the outer or inner term (whichever one the GCF is next to) Row 1: has an ‘x’ in common Column 1: an ‘a’ in common Row 2: ‘y’ in common Column 2: a ‘2b’ in common

+y - 2b ( ) Answer: (x + y)(a - 2b)
The terms on the outside of the box are the answer. Write parentheses around each pair and place them side by side as a product. ( ) - 2b F O I L +y This whole procedure takes about 4 steps, it only looks longer in this presentation because I have explained each step in detail. Here are the steps: 1. Draw a two by two box 2. Place the terms inside the box in the correct location 3. Factor out the GCF for each row and column 4. Write these terms in the form of a product Answer: (x + y)(a - 2b)

Notice, when there seems to be nothing in
Example 2: 2xy + y - 4x - 2 Draw the box and place the terms in the correct spaces 2x + 1 Factor out the GCF for each row & column Notice, when there seems to be nothing in common, we take out a ‘1’. y 2xy + y - 2 - 4x - 2 When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. Answer: (y - 2)(2x + 1)

SPECIAL CASE!!! There are some polynomials that have a factor that must be taken out before using the box. Take a polynomial such as: 4xy + 2y - 8x - 4 This polynomial has a common factor of “2” in all four terms. If this is not taken out before using the box, the “2” will be taken out twice, doubling the answer. (Example on the next two slides)

The way this problem should NOT be worked 4xy + 2y - 8x - 4 This problem has a GCF of “2” that wasn’t factored out! 4x + 2 The box “SEES” the “2” in both dimensions and pulls it out twice!!! If we check the answer using FOIL, we DO NOT get the original problem!!! 2y 4xy + 2y - 4 - 8x - 4 When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. (2y - 4)(4x + 2)= 8xy + 4y - 16x - 8 X WRONG ANSWER!!!!!!

2xy + y - 4x - 2 2x + 1 y - 2 2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4 
The way this problem should have been worked 4xy + 2y - 8x - 4 Example: This problem has a GCF of “2”, so factor it out FIRST! 2 (2xy + y - 4x - 2) 2x + 1 Now, place the four terms into the box and factor normally y 2xy Don’t forget to put that extra “2” in the answer!!! + y When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. - 2 - 4x - 2 2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4 

Note #4 take out the GCF before factoring
Some practice problems 1. 2. 3. 4. 2xy - 6x + 4y -12 Note #4 take out the GCF before factoring Here are some practice problems with the answers at the bottom, so you can try out this box for yourself Answers: 1. (a + b)(x- y) , 2. (a - b)(5x - 2y), 3.(a - 2)(x2 + 3), 4. 2(x + 2)(y - 3)

This is called “splitting the middle term”.
The Greenebox factoring method uses a box with four spaces in it. That means it works on polynomials that have 4 terms. In other words: A polynomial with 3-terms can only be put into the box after we change the three terms back into the original four. This is called “splitting the middle term”.

Step 1: Multiply first*last Step 4: Which pair of factors gives us
2x2 + 9x + 10 first middle last Splitting the middle term Step 1: Multiply first*last 2 x 10 = 20 Step 3: The sign of the last term tells us whether to add or subtract the factors of 20. Add 20 Step 2: Find all the FACTORS of 20 1  20 2  10 4  5 = 21 = 12 = 9 Step 4: Which pair of factors gives us the middle term? Since the middle term is 9x, the original 4 and 5 each had an“x”. (They were like terms and were added together) This is the step-by-step method of splitting the middle term. Again, this is a lot shorter when you actually use it. 1. Multiply the first times the last term 2. Find all the factors of the product 3. Look at the sign of the last term to see if you should add or subtract these pairs of factors 4. Pick the factors that give you the correct middle term 5. Choose the correct signs: If you added the factors, both factors have the sign of the middle term 6. If you subtracted the factors, one will be positive and the other negative (The bigger number will have the same sign as the middle term) IMPORTANT NOTE: SINCE THE TWO FACTORS WE HAVE CHOSEN ADD UP TO THE MIDDLE TERM WHICH HAS AN ‘x’ IN IT, THEY EACH ALSO HAVE AN ‘x’. + 4x and + 5x = + 9x This pair gives us the correct middle term.

1. Split the middle term into 2 terms
Note: We did step 1 on the previous page 2x2 + 9x + 10 1. Split the middle term into 2 terms F O I L 2x2 + 4x + 5x + 10 2. Place each term in the correct location in the box x + 2 F O I L 2x 2x2 + 4x 3. Factor out GCF for each row & column 4. Answer: (x + 2)(2x + 5) Now that the polynomial has been returned to it’s original four term status, we can factor it using the box 1. Draw the box 2. Place the 4 terms in the correct locations, with their signs * Remember that the two middle terms can go in either position, but the first and last terms must go on the top left and bottom right, respectively* 3. Factor out the GCF for each row and column, taking the sign of the middle terms. 4. Write these in the form of a product +5 + 5x + 10

Step 3: Pick the pair that subtract to equal -10x
Subtraction example first last Step 1: Multiply first*last 3x2 - 10x - 8 SUBTRACT 3 x 8 = 24 24 Step 2. Find all the factors of 24 1 * 24 2 * 12 3 * 8 4 * 6 = 23 = 10 = 5 = 2 Step 3: Pick the pair that subtract to equal -10x (the middle term) This pair works Step 4: Pick the correct signs: (Subtract means: different signs) +12x - 2x = +10x - 12x + 2x = -10x Here is an example of the same process but the last term is negative 1. This page shows how to split the middle term when subtracting correct terms: -12x and +2x

3. Factor out GCF for each row & column
3x2 - 10x - 8 1.Split the middle term into 2 terms F O I L 3x2 + 2x - 12x - 8 2. Place each term in the correct location in the box 3x +2 F O I L x 3x2 +2x 3. Factor out GCF for each row & column 4. Answer: (3x + 2)(x - 4) 2. Draw the box 3. Place the four terms in the correct location inside the box 4. Factor out the GCF, taking the signs from the middle terms 5. Write the answer in the form of a product -4 - 8 - 12x

x2 +3x - 10 - Note:first= 1 Now factor it -2x and +5x = +3x
II. Factoring a Trinomial of the form: x2 + bx - c Note:first= 1 x2 +3x - 10 - Step 3: Subtract Step 1: Multiply the first*last 1 * 10 = 10 1 * 10 2 * 5 = 9 = 3 Step 2: Find the factors of 10 Step 4: Choose the pair of factors that equal + 3x (the middle term) Step 5: Choose the correct signs: -2x and +5x = +3x or -5x and +2x = -3x Usually, after letting the students work about three of these problems, I point out to them that if they use only the last term, they get the same product When subtracting, the signs will be different (One “+” , the other “-”) Now factor it

Factor out GCF for each row & column
x2 + 3x - 10 Split the middle term into 2 terms F O I L x2 + 5x - 2x - 10 Place each term in the correct location in the box x +5 F O I L x 2 x + 5x Factor out GCF for each row & column Answer: (x + 5)(x - 2) I then point out to them that the numbers that they put into the box (in the outer and inner slots) are the same ones they eventually factored out. So, if they take those two factors and place them directly into the answer, they can bypass the use of the box. (ONLY IF THE COEFFICIENT OF THE FIRST TERM IS A 1) - 2 - 2x - 10 Note: Once you’ve found the split terms and the signs, you can go straight to the answer.

Split this one on your own:
4x2 + 10x - 6 4x2 + 12x - 2x - 6 2x + 6 If we check the answer using FOIL, we DO NOT get the original problem!!! 4x2 + 10x - 6 4x 4x2 + 12x - 2 - 2x - 6 When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. (2x + 6)(4x - 2)= 8x2 + 20x - 12X WRONG ANSWER!!!!!! THIS MEANS THERE WAS A FACTOR WE MISSED!!!

Step 4: Choose the pair of factors that equal + 5x
The way this problem SHOULD be worked This problem has a GCF of “2”. We’ll factor it out then work the problem normally using only the trinomial (in parentheses). - Step 3: Subtract 4x2 + 10x - 6 = 2 (2x2 + 5x - 3) Step 1: Multiply the first*last 2 * 3 = 6 1 * 6 2 * 3 Step 2: Find the factors of 6 = 5 = 1 Step 4: Choose the pair of factors that equal + 5x (the middle term) Step 5: Choose the correct signs: -x and +6x = +5x or x and -6x = -5x

2x2 + 6x - x - 3 x + 3 2x - 1 2(x + 3)(2x - 1)= 4x2 + 10x - 6 
The way this problem should have been worked 2 (2x2 + 6x - x - 3) Example: x + 3 Now, place the four terms into the box and factor normally 2x2 2x + 6x Don’t forget to put that extra “2” in the answer!!! - 1 - x - 3 When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. 2(x + 3)(2x - 1)= 4x2 + 10x - 6 

4x2 - 5x - 6 3y2 - 16y + 5 2x2 + 9x - 18 Answers: 1. (4x + 3)(x - 2)
Some practice problems Answers: 1. (4x + 3)(x - 2) 2. (3y - 1)(y - 5) 3.(2x - 3)(x + 6) 4. (x + 1)(x + 2) 5. (x - 6)(x + 2) 4x2 - 5x - 6 1. 2. 3. 3y2 - 16y + 5 2x2 + 9x - 18 4. x2 + 3x + 2 5. x2 - 4x - 12 Make sure to check for overall GCF’s before using the box.

-1 (3x2 - 4x + 5) This changes all the signs! Important note:
Just as in regular methods for factoring, the first term must be a positive number. If it is not, then factor out a ‘-1’ . Example: x2 + 4x - 5 This negative must be removed (factored-out) -1 (3x2 - 4x + 5) This changes all the signs!

x2 - 4 x2 + 0x - 4 The difference of two squares +2x - 2x = 0x
FIRST LAST Rewrite as a Trinomial like this: 1. First * Last 1 * 4 = 4 x2 + 0x - 4 2. Find the factors of 4 1*4 2*2 +2x - 2x = 0x I have included this example to demonstrate how the box CAN be used to factor the difference of two squares. However, it is mostly academic (to show that it can be done). I encourage my own students to simply use the formula because it is faster. Now factor it 3. To equal 0x, the terms must be: the same number with opposite signs.

x - 0x - 4 x - 2x + 2x - 4 -2 x + 2 x x + 2x - 4 - 2x 2 Answer:
Split the middle term into 2 terms x F O I L 2 - 2x + 2x - 4 Place each term in the correct location in the box x + 2 F I O L x x 2 + 2x Factor out the GCF for each row & column Answer: (x + 2)(x - 2) -2 - 2x - 4 15 Note: It’s much shorter to use the difference of two squares formula 17

- 3x - 3 2x2 - 3 x 2x3 +1 + 2x2 Answer: (2x2 - 3)(x + 1)
A very few CUBIC POLYNOMIALS can be factored using the Greenebox Method. **You must check the answer** Example: 2x3 + 2x2 - 3x - 3 Draw the box and place the terms in the correct spaces 2x2 - 3 Factor out the GCF for each row & column x - 3x 2x3 When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term. +1 + 2x2 - 3 Answer: (2x2 - 3)(x + 1)