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The Greenebox The Greenebox Factoring MethodFactoring Method

Copyright 1999 Lynda Greene all rights reserved

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F O I L

ax bx ay by 2 24 terms:

Example 1:

FIRST STEP: Draw a box and insert the four terms in the correctpositions. Use the RED (FOIL) letters as your guide.

F O

I L

ax

ay

bx2

2by

(include thesign “+, -” tothe left of eachterm)

Greenebox Factoring Method- Four TermsGreenebox Factoring Method- Four Terms

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This is called the GCF(Greatest Common Factor)

Write the GCF next to each row or column

Row 1: has an ‘x’ in common

x

Row 2: ‘y’ in common

y

Column 1: an ‘a’ in common

a

Column 2: a ‘2b’ in common

2b

Take the signs of the outer or inner terms:O: O: take ‘-’ from -2bx I: I: and ‘+’ from +ay+

-

ax

2byI L

F O

+ay

-2bx

Looking at terms two at a time, what does each pair have in common?

ASK YOURSELFASK YOURSELF::

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ax

ay

bx2

2byI L

F O

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.

( )

(

)

Answer: (x + y)(a - 2b)

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2xy + y - 4x - 2Example 2:

Draw the box and place the terms in the correct spaces

2xy + y

- 4x - 2

Factor outthe GCFfor each row &column

y

+ 1 Notice, when

there seems to be nothing incommon, we

take out a ‘1’.

2x

- 2

Answer: (y - 2)(2x + 1)Answer: (y - 2)(2x + 1)

SPECIAL CASE!!!There are some polynomials that have a factor that

must be taken out beforebefore 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)

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4xy + 2y - 8x - 4This problem has a GCF of “2” that wasn’t factored out!

4xy + 2y

- 8x - 4

The box “SEES”the “2”in both

dimensionsand pulls itout twice!!!

2y

+ 2 If we checkthe answer

using FOIL, weDO NOT getthe original problem!!!

4x

- 4

(2y - 4)(4x + 2)= 8xy + 4y - 16x - 8 (2y - 4)(4x + 2)= 8xy + 4y - 16x - 8 XXWRONG ANSWER!!!!!!WRONG ANSWER!!!!!!

The way this problem should NOT be workedThe way this problem should NOT be worked

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4xy + 2y - 8x - 4Example:

This problem has a GCF of “2”, so factor it out FIRST!

2xy + y

- 4x - 2

Now, placethe four terms

into the boxand factor normally

y

+ 1 Don’t forget toput that extra

“2” in the answer!!!

2x

- 2

2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4 2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4

The way this problem should have been workedThe way this problem should have been worked

2 (2xy + y - 4x - 2)

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Some practice problems

ax ay bx by 1.

2. 5 5 2 2ax bx ay by

3. ax x a2 22 3 6

4. 2xy - 6x + 4y -12 Note #4 take out the GCF before factoring

Answers: 1. (a + b)(x- y) , 2. (a - b)(5x - 2y), 3.(a - 2)(x2 + 3), 4. 2(x + 2)(y - 3)

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SPLITTING THE MIDDLE TERMSPLITTING THE MIDDLE TERM

The The Greenebox factoring methodGreenebox factoring method uses a uses a box with four spaces in it. That means it box with four spaces in it. That means it works on polynomials that have 4 terms. works on polynomials that have 4 terms.

A polynomial with 3-terms can only be put into the box after we change the three terms change the three terms

back into the original fourback into the original four. This is called “splitting the middle term”.

In other words:In other words:

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Step 1: Multiply first*last

2 x 10 = 20

Step 2: Find all the FACTORS of 20

201 202 104 5

Step 3: The sign of the last term tells us whether to add or

subtract the factors of 20.

Add= 21= 12= 9

Step 4: Which pair of factors gives us the middle term?

+ 4x and + 5x = + 9x + 4x and + 5x = + 9x This pair gives us the correct middle term.This pair gives us the correct middle term.

2x2 + 9x + 10

first middle lastSplitting 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)

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F O I L 2x2 + 4x + 5x + 10

Note: We did step 1 on the previous page

2. Place each term in the correct location in the box

F O

I L

2x2

+ 5x

+ 4x

+ 10

3. Factorout GCF for each row &column

2x

+5

x

2x2 + 9x + 10

+ 2

4. Answer:4. Answer: (x + 2)(2x + 5)(x + 2)(2x + 5)

1. Split the middle term into 2 terms

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Step 1: Multiply first*last

3x2 - 10x - 8first last

3 x 8 = 24

Step 2. Find all the factors of 24 24

SubtractionSubtraction exampleexample

1 * 242 * 123 * 84 * 6

Step 3: Pick the pair that subtract to equal -10x

(the middle term)

SUBTRACT = 23 = 10 = 5 = 2

This pair worksStep 4: Pick the correct signs:(Subtract means: different signs) +12x - 2x = +10x - 12x + 2x = -10x

correct terms: -12x and +2x

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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

F O

I L

3x2

- 12x

+2x

- 8

3. Factor out GCF for each row & column

x

-4

3x +2

4. Answer: (3x + 2)(x - 4)

3x2 - 10x - 8

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x2 +3x - 10

II. Factoring a Trinomial of the form: xII. Factoring a Trinomial of the form: x2 2 + bx - c+ bx - c

Step 1: Multiply the first*last

Note:first= Note:first= 11 1 * 10 = 10

Step 2: Find the factors of 10 1 * 102 * 5

- Step 3: Subtract= 9= 9= 3= 3

Step 4: Choose the pair of factors that equal + 3x (the middle term)

Step 5: Choose the correct signs:

-2x and +5x = +3xor -5x and +2x = -3xWhen subtracting, the signs

will be different (One “+” , the other “-”)

Now factor it Now factor it

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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

F O

I L

2 x + 5x

- 2x - 10

Factor out GCF for

each row & column

x

+5 x

- 2

Answer: (x + 5)(x - 2)

Note: Once you’ve found the split terms and the signs, you can go straight to the answer.

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4x2 + 12x

- 2x - 6

4x

+ 6 If we checkthe answer

using FOIL, weDO NOT getthe original problem!!!

4x2 + 10x - 6

2x

- 2

(2x + 6)(4x - 2)= 8x(2x + 6)(4x - 2)= 8x22 + 20x - 12 + 20x - 12XXWRONG ANSWER!!!!!! WRONG ANSWER!!!!!!

THIS MEANS THERE WAS A FACTOR WE MISSED!!!THIS MEANS THERE WAS A FACTOR WE MISSED!!!

4x2 + 12x - 2x - 64x2 + 10x - 6

Split this one on your own:

4x2 + 10x - 6 = 2 (2x2 + 5x - 3)

This problem has a GCF of “2”. We’ll factor it out then work the problem normally using only the trinomial (in parentheses).

The way this problem SHOULD be workedThe way this problem SHOULD be worked

Step 1: Multiply the first*last

2 * 3 = 6Step 2: Find the factors of 6

1 * 62 * 3

- Step 3: Subtract

= 5= 5= 1= 1

Step 4: Choose the pair of factors that equal + 5x

(the middle term)Step 5: Choose the correct signs:

-x and +6x = +5xor x and -6x = -5x

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2 (2x2 + 6x - x - 3)Example:

2x2 + 6x

- x - 3

Now, placethe four terms

into the boxand factor normally

2x

+ 3 Don’t forget toput that extra

“2” in the answer!!!

x

- 1

2(x + 3)(2x - 1)= 4x2(x + 3)(2x - 1)= 4x22 + 10x - 6 + 10x - 6

The way this problem should have been workedThe way this problem should have been worked

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Some practice problems

1.

2.

3.

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

3y2 - 16y + 5

2x2 + 9x - 18

4. x2 + 3x + 2

5. x2 - 4x - 12

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: -3x2 + 4x - 5

This negative must be removed (factored-out)

-1 (3x2 - 4x + 5)This changes all the signs!

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The difference of two squaresThe difference of two squares

x2 - 4FIRST LAST

Rewrite asa Trinomiallike this:x2 + 0x - 4

1. First * Last 1 * 4 = 4

2. Find the factors of 4 1*4 2*2

3. To equal 0x, the terms must be: the same number with opposite signs.

+2x - 2x = 0x

Now factor itNow factor it

2317

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2 x - 0x - 4 Split the middle term into 2 terms

xF O I L2 - 2x+ 2x - 4

Place each term in the correct location in the box

F I

O L

x2

- 2x

+ 2x

- 4

Factor outthe GCFfor eachrow &column

x

-2

x + 2

Answer: (x + 2)(x - 2)

Note: It’s much shorter to use the difference of two squares formula

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Example: 2x3 + 2x2 - 3x - 3

A very few CUBIC POLYNOMIALSCUBIC POLYNOMIALS can be factored using the Greenebox Method. **You must check the answer**

Draw the box and place the terms in the correct spaces

2x3

+ 2x2

- 3x

- 3

Factor outthe GCFfor each row &column

x

+1

2x2 - 3

Answer: (2xAnswer: (2x22 - 3)(x + 1) - 3)(x + 1)