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1 The Greenebox Factoring Method Copyright 1999 Lynda Greene all rights reserved

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2 F O IL 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. FO IL (include the sign “+, -” to the left of each term) Greenebox Factoring Method- Four Terms

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3 This is called the GCF (Greatest Common Factor) Write the GCF next to each row or column Row 1: has an ‘x’ in common Row 2: ‘y’ in common y Column 1: an ‘a’ in common 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 + - IL F O +ay -2bx Looking at terms two at a time, what does each pair have in common? ASK YOURSELF:

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4 IL FO +y - 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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5 2xy + y - 4x - 2 Example 2: Draw the box and place the terms in the correct spaces 2xy + y - 4x - 2 Factor out the GCF for each row & column y + 1 Notice, when there seems to be nothing in common, we take out a ‘1’. 2x - 2 Answer: (y - 2)(2x + 1)

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

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7 4xy + 2y - 8x - 4 This problem has a GCF of “2” that wasn’t factored out! 4xy + 2y - 8x - 4 The box “SEES” the “2” in both dimensions and pulls it out twice!!! 2y + 2 If we check the answer using FOIL, we DO NOT get the original problem!!! 4x - 4 (2y - 4)(4x + 2)= 8xy + 4y - 16x - 8 X WRONG ANSWER!!!!!! The way this problem should NOT be worked

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8 4xy + 2y - 8x - 4 Example: This problem has a GCF of “2”, so factor it out FIRST! 2xy + y - 4x - 2 Now, place the four terms into the box and factor normally y + 1 Don’t forget to put that extra “2” in the answer!!! 2x - 2 2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4 The way this problem should have been worked 2 (2xy + y - 4x - 2)

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9 Some practice problems 1. 2. 3. 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)(x 2 + 3), 4. 2(x + 2)(y - 3)

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10 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. change the three terms back into the original four 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”. In other words:

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11 Step 1: Multiply first*last 2 x 10 = 20 Step 2: Find all the FACTORS of 20 20 1 20 2 10 4 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 This pair gives us the correct middle term. 2x 2 + 9x + 10 first middle last Splitting 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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12 FOIL 2x 2 + 4x + 5x + 10 Note: We did step 1 on the previous page 2. Place each term in the correct location in the box FO IL 2x 2 + 5x + 4x + 10 3. Factor out GCF for each row & column 2x +5 x 2x 2 + 9x + 10 + 2 4. Answer: (x + 2)(2x + 5) (x + 2)(2x + 5) 1. Split the middle term into 2 terms

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13 Step 1: Multiply first*last 3x 2 - 10x - 8 first last 3 x 8 = 24 Step 2. Find all the factors of 24 24Subtraction example example 1 * 24 2 * 12 3 * 8 4 * 6 Step 3: Pick the pair that subtract to equal -10x (the middle term) SUBTRACT = 23 = 10 = 5 = 2 This pair works Step 4: Pick the correct signs: (Subtract means: different signs) +12x - 2x = +10x - 12x + 2x = -10x correct terms: -12x and +2x

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14 1.Split the middle term into 2 terms FOIL 3x 2 + 2x - 12x - 8 2. Place each term in the correct location in the box FO IL 3x 2 - 12x +2x - 8 3. Factor out GCF for each row & column x -4 3x +2 4. Answer: (3x + 2)(x - 4) 3x 2 - 10x - 8

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15 x2 x2 +3x - 10 II. Factoring a Trinomial of the form: x 2 + bx - c Step 1: Multiply the first*last Note:first= 1 1 * 10 = 10 Step 2: Find the factors of 10 1 * 10 2 * 5 - Step 3: Subtract = 9 = 3 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 When subtracting, the signs will be different (One “+”, the other “-”) Now factor it

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16 x 2 + 3x - 10 Split the middle term into 2 terms FOIL x 2 + 5x - 2x - 10 Place each term in the correct location in the box FO IL 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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17 4x 2 + 12x - 2x - 6 4x + 6 If we check the answer using FOIL, we DO NOT get the original problem!!! 4x 2 + 10x - 6 2x - 2 (2x + 6)(4x - 2)= 8x2 + 20x - 12X WRONG ANSWER!!!!!! THIS MEANS THERE WAS A FACTOR WE MISSED!!! 4x 2 + 12x - 2x - 6 4x 2 + 10x - 6 Split this one on your own:

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4x 2 + 10x - 6 = 2 (2x 2 + 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 worked Step 1: Multiply the first*last 2 * 3 = 6 Step 2: Find the factors of 6 1 * 6 2 * 3 - Step 3: Subtract = 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

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19 2 (2x 2 + 6x - x - 3) Example: 2x 2 + 6x - x - 3 Now, place the four terms into the box and factor normally 2x + 3 Don’t forget to put that extra “2” in the answer!!! x - 1 2(x + 3)(2x - 1)= 4x2 + 10x - 6 The way this problem should have been worked

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20 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) 4x 2 - 5x - 6 3y 2 - 16y + 5 2x 2 + 9x - 18 4. x 2 + 3x + 2 5. x 2 - 4x - 12

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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: -3x 2 + 4x - 5 This negative must be removed (factored-out) -1 (3x 2 - 4x + 5) This changes all the signs!

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22 The difference of two squares x 2 - 4 FIRST LAST Rewrite as a Trinomial like this: x 2 + 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 it

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23 17 15 2 x - 0x - 4 Split the middle term into 2 terms x FOIL 2 - 2x+ 2x - 4 Place each term in the correct location in the box FI OL x 2 - 2x + - 4 Factor out the GCF for each row & 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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24 Example: 2x 3 + 2x 2 - 3x - 3 CUBIC POLYNOMIALS A very few CUBIC POLYNOMIALS can be factored using the Greenebox Method. **You must check the answer** Draw the box and place the terms in the correct spaces 2x 3 + 2x 2 - 3x - 3 Factor out the GCF for each row & column x +1 2x 2 - 3 Answer: (2x2 - 3)(x + 1)

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