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

3 Special Cases A. Always take out GCF B. Difference of 2 squares C. Some 4-term cubics

4 Warning: Before placing terms into the box you must first factor out (remove) the GCF you must first factor out (remove) the GCF if it exists Example: 4x x 2 + 8x This polynomial has a common factor of 2x 1) Factor out GCF: 2x (2x 2 - 9x + 4) Now, use the Greenebox Method to factor the trinomial (in parentheses)

5 2x (2x 2 - 9x + 4) Multiply the first and last numbers 2 * 4 = 8 Find the factors of 8 that add up to -9x 8 1 * 8 2 * 4 Add = 9 = 6 -1x and -8x equal -9x 1x and 8x have the same sign (-) Now put 2x 2 -1x - 8x + 4 into the box & factor, we will put the 2x back in front when we finish.

6 Split the middle term into 2 terms FOIL 2x(2x 2 - 1x - 8x + 4) Place each term in the correct location in the box FI OL 2x 2 - 1x - 8x + 4 Factor out GCF for each row & column 2x x - 4 Answer: 2x(2x - 1)(x - 4) Don’t forget the GCF 2x( 2 x 2 - 9x + 4)

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!

8 Some practice problems Answers: 1) 2(4x + 3)(x - 2) 2) y(3y - 1)(y - 5) 3) 3x(2x - 3)(x + 6) 4) 4(x + 1)(x + 2) 5) y(x - 6)(x + 2) 6) 2z(x - 3)(x + 6) 8x x y y 2 + 5y -4x x x 4. 4 x x x 2 y - 4xy - 12y 6. 2x 2 z + 6xz - 36z

9 The difference of two squares x FIRST LAST Rewrite as a Trinomial like this: x 2 + 0x 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

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 Factor out the GCF for each row & column x -2 x+ 2 Answer: (x + 2)(x - 2)

11 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 Answer: (2x2 - 3)(x + 1)

12 Summary: The Greenebox Factoring Method I. Factoring 4 term polynomials: Identify the first, outer, inner, last terms (FOIL) Draw the box and place terms in the correct places Factor out the GCF for each row and column Take the signs of the outer and inner term II. Factoring Trinomials of the form: ax 2 + bx + c Multiply the first and last term (coefficients only) List the factors of this product Find the factors that add/subtract to equal the middle term The sign of the last term tells you whether to add or subtract the factors Choose the correct signs for the two terms If you added the factors, they have the same signs If you subtracted, they have different signs they should equal the middle term Split the middle term into outer and inner terms and use the factoring procedure listed in part I

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