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SLIDE SHOW INSTRUCTIONS This presentation is completely under your control. This lesson will show only one step at a time, to see the next step you must press a key. to see the next step you must press a key. (Actual names written on a key are in green) TO STOP THE SLIDE SHOW: press ‘escape’ (Esc, top left of keyboard) TO MOVE FORWARD: press the “spacebar” or Enter (PageDn, , , also work) TO MOVE BACKWARD: press the key (PageUp, or also work)

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

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3 Factoring Trinomials: 1. Split the middle term 1. Split the middle term 2. Factor using the box 2. Factor using the box Factoring Trinomials: 1. Split the middle term 1. Split the middle term 2. Factor using the box 2. Factor using the box

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4 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. A polynomial with 4-terms can be put into the box and factored immediately. change the three terms back into the original four A polynomial with 3-terms can 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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Why do some polynomials have 3 terms and others have 4? When the FOIL multiplication method is used to multiply 2 binomials, for example: (x + 3)(4x - 2) or (a + b)(c - d) If the Outer and Inner terms can be combined, the result will be a Trinomial (3 terms). (x + 3)(4x - 2)= x 2 - 2x + 12x - 6 = x x - 6 If they cannot be combined, it keeps all 4 terms. (a + b)(c - d) = ac - ad + bc - bd Each answer will have four terms (First, Outer, Inner, Last) An earlier lesson covered the process for factoring 4-terms, this tutorial covers factoring 3-term polynomials.

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6 II. Factoring a Trinomial of the form: ax 2 + bx + c 3 terms: 2x 2 + 9x + 10 First (a) Middle (b) Last (c) First, we will split the middle term into the original outer and inner terms. This gives us a four term polynomial which we can factor using the Greenebox method.

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7 Step 1: Multiply first*last 2 x 10 = 20 Step 2: Find all the FACTORS of 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? Step 5: Choose the correct signs. FACT: We add numbers when they have the same signs. (step 3). both positiveboth negative i.e. The factors are both positive, + 4x and + 5x OR both negative, -4x and -5x + 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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8 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 Factor out GCF for each row & column 2x +5 x 2x 2 + 9x Answer: (x + 2)(2x + 5) (x + 2)(2x + 5) 1. Split the middle term into 2 terms

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

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11 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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12 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)

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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: 4x x - 6 This polynomial has a common factor of “2” in all three 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) This is a useful feature of the Greenebox Method. It forces the student to take out the GCF BEFORE trying to factor the terms by catching extra factors which might otherwise go unnoticed. If an extra factor is not taken out at some point, the answer will be wrong.

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4x x - 6 This problem has a GCF of “2”. We’ll pretend we didn’t see it and work the problem normally, The way this problem should NOT be worked Step 1: Multiply the first*last 4 * 6 = 24 Step 2: Find the factors of 24 1 * 24 2 * 12 3 * 8 4 * 6 - Step 3: Subtract = 23 = 10 = 5 = 2 Step 4: Choose the pair of factors that equal + 10x (the middle term) Step 5: Choose the correct signs: - 2x and +12x = +10x or 2x and -12x = -10x

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15 Put the four factors into the box and factor it 4x x - 2x - 6 The box “SEES” the “2” in both dimensions and pulls it out twice!!! 4x + 6 If we check the answer using FOIL, we DO NOT get the original problem!!! 4x x - 6 2x - 2 (2x + 6)(4x - 2)= 8x2 + 20x - 12X WRONG ANSWER!!!!!! THIS MEANS THERE WAS A FACTOR WE MISSED!!! The way this problem should NOT be worked 4x x - 2x - 6

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4x x - 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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17 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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18 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) 4x 2 - 5x - 6 3y y + 5 2x 2 + 9x x 2 + 3x x 2 - 4x - 12

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