Presentation on theme: "The Greenebox Factoring Method"— Presentation transcript:
1 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 Greeneall rights reserved
2 Greenebox Factoring Method- Four Terms L4 terms:Example 1:FIRST STEP: Draw a box and insert the four terms in the correctpositions. Use the RED (FOIL) letters as your guide.FOIL(include thesign “+, -” tothe left of eachterm)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.
3 +ay - 2b -2bx y + ASK YOURSELF: Looking at terms two at a time, what does each pair have in common?-2bThis is called the GCF(Greatest Common Factor)Write the GCF next toeach row or columnOF-2bxTake the signs of the outer or inner terms:O: take ‘-’ from -2bxI: and ‘+’ from +ayIL+ay+yRecall that the GCF is the largest factor each of the terms has in commonin a particular row or column.Example: the two terms in the first row each have an ‘a’ in commonThe sign is always taken from the outer or inner term (whichever one the GCF is next to)Row 1: has an ‘x’ in commonColumn 1: an ‘a’ in commonRow 2: ‘y’ in commonColumn 2: a ‘2b’ in common
4 +y - 2b ( ) Answer: (x + y)(a - 2b) The terms on the outside of the box are the answer.Write parentheses around each pairand place them side by side as a product.()- 2bFOIL+yThis 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 box2. Place the terms inside the box in the correct location3. Factor out the GCF for each row and column4. Write these terms in the form of a productAnswer: (x + y)(a - 2b)
5 Notice, when there seems to be nothing in Example 2:2xy + y - 4x - 2Draw the box and place the terms in the correct spaces2x+ 1Factor outthe GCFfor eachrow &columnNotice, when there seems to be nothing incommon, we take out a ‘1’.y2xy+ y- 2- 4x- 2When 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)
6 SPECIAL CASE!!!There are some polynomials that have a factor thatmust be taken out before using the box.Take a polynomial such as:4xy + 2y - 8x - 4This 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)
7 the answer using FOIL, we The way this problem should NOT be worked4xy + 2y - 8x - 4This problem has a GCF of “2” that wasn’t factored out!4x+ 2The box“SEES”the “2”in bothdimensionsand pulls itout twice!!!If we checkthe answer using FOIL, weDO NOT getthe original problem!!!2y4xy+ 2y- 4- 8x- 4When 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 XWRONG ANSWER!!!!!!
8 2xy + y - 4x - 2 2x + 1 y - 2 2(y - 2)(2x + 1)= 4xy + 2y - 8x - 4 The way this problem should have been worked4xy + 2y - 8x - 4Example:This problem has a GCF of “2”, so factor it out FIRST!2 (2xy + y - 4x - 2)2x+ 1Now, placethe four termsinto the boxand factornormallyy2xyDon’t forget toput that extra“2” in the answer!!!+ yWhen it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term.- 2- 4x- 22(y - 2)(2x + 1)= 4xy + 2y - 8x - 4
9 Note #4 take out the GCF before factoring Some practice problems126.96.36.199. 2xy - 6x + 4y -12Note #4 take out the GCF before factoringHere are some practice problems with the answers at the bottom, so you can try out this box for yourselfAnswers: 1. (a + b)(x- y) , 2. (a - b)(5x - 2y), 3.(a - 2)(x2 + 3),4. 2(x + 2)(y - 3)
10 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”.
11 Step 1: Multiply first*last Step 4: Which pair of factors gives us 2x2 + 9x + 10firstmiddlelastSplitting the middle termStep 1: Multiply first*last2 x 10 = 20Step 3: The sign of the last term tells us whether to add or subtract the factors of 20.Add20Step 2: Find all theFACTORS of 201 202 104 5= 21= 12= 9Step 4: Which pair of factors gives usthe 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 term2. Find all the factors of the product3. Look at the sign of the last term to see if you should add or subtract these pairs of factors4. Pick the factors that give you the correct middle term5. Choose the correct signs: If you added the factors, both factors have the sign of the middle term6. 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 = + 9xThis pair gives us the correct middle term.
12 1. Split the middle term into 2 terms Note: We did step 1 on the previous page2x2 + 9x + 101. Split the middle term into 2 termsFOIL2x2 + 4x + 5x + 102. Place each term in the correct location in the boxx+ 2FOIL2x2x2+ 4x3. Factorout GCFfor eachrow &column4. 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 box1. Draw the box2. 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
13 Step 3: Pick the pair that subtract to equal -10x SubtractionexamplefirstlastStep 1: Multiply first*last3x2 - 10x - 8SUBTRACT3 x 8 = 2424Step 2. Find all the factors of 241 * 242 * 123 * 84 * 6= 23= 10= 5= 2Step 3: Pick the pair that subtract to equal -10x(the middle term)This pair worksStep 4: Pick the correct signs:(Subtract means: different signs)+12x - 2x = +10x- 12x + 2x = -10xHere is an example of the same process but the last term is negative1. This page shows how to split the middle term when subtractingcorrect terms: -12x and +2x
14 3. Factor out GCF for each row & column 3x2 - 10x - 81.Split the middleterm into 2 termsFOIL3x2 + 2x - 12x - 82. Place each term in the correct location in the box3x+2FOILx3x2+2x3. Factor out GCF for each row & column4. Answer:(3x + 2)(x - 4)2. Draw the box3. Place the four terms in the correct location inside the box4. Factor out the GCF, taking the signs from the middle terms5. Write the answer in the form of a product-4- 8- 12x
15 x2 +3x - 10 - Note:first= 1 Now factor it -2x and +5x = +3x II. Factoring a Trinomial of the form: x2 + bx - cNote:first= 1x2 +3x - 10-Step 3:SubtractStep 1: Multiplythe first*last1 * 10 = 101 * 102 * 5= 9= 3Step 2: Find the factors of 10Step 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 = -3xUsually, 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 productWhen subtracting, the signs will be different(One “+” , the other “-”)Now factor it
16 Factor out GCF for each row & column x2+ 3x- 10Split the middleterm into 2 termsFOILx2+ 5x- 2x- 10Place each term in the correct location in the boxx+5FOILx2x+ 5xFactor out GCF for each row & columnAnswer:(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- 10Note: Once you’ve found the split terms and the signs,you can go straight to the answer.
17 Split this one on your own: 4x2 + 10x - 64x2 + 12x - 2x - 62x+ 6If we checkthe answer using FOIL, weDO NOT getthe original problem!!!4x2 + 10x - 64x4x2+ 12x- 2- 2x- 6When 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 - 12XWRONG ANSWER!!!!!! THIS MEANS THERE WAS A FACTOR WE MISSED!!!
18 Step 4: Choose the pair of factors that equal + 5x The way this problem SHOULD be workedThis problem has a GCF of “2”. We’ll factor it out then work the problem normally using only the trinomial (in parentheses).-Step 3:Subtract4x2 + 10x - 6 = 2 (2x2 + 5x - 3)Step 1: Multiplythe first*last2 * 3 = 61 * 62 * 3Step 2: Find the factors of 6= 5= 1Step 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
19 2x2 + 6x - x - 3 x + 3 2x - 1 2(x + 3)(2x - 1)= 4x2 + 10x - 6 The way this problem should have been worked2 (2x2 + 6x - x - 3)Example:x+ 3Now, placethe four termsinto the boxand factornormally2x22x+ 6xDon’t forget toput that extra“2” in the answer!!!- 1- x- 3When 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
21 -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 - 5This negative must be removed (factored-out)-1 (3x2 - 4x + 5)This changes all the signs!
22 x2 - 4 x2 + 0x - 4 The difference of two squares +2x - 2x = 0x FIRST LASTRewrite asa Trinomiallike this:1. First * Last1 * 4 = 4x2 + 0x - 42. Find the factors of 41*42*2+2x - 2x = 0xI 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 it3. To equal 0x, the terms must be:the same number with opposite signs.
23 x - 0x - 4 x - 2x + 2x - 4 -2 x + 2 x x + 2x - 4 - 2x 2 Answer: Split the middleterm into 2 termsxFOIL2- 2x+ 2x- 4Place each term in the correct location in the boxx+ 2FIOLxx2+ 2xFactor outthe GCFfor eachrow &columnAnswer:(x + 2)(x - 2)-2- 2x- 415Note: It’s much shorter to use the difference of two squares formula17
24 - 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 - 3Draw the box and place the terms in the correct spaces2x2- 3Factor outthe GCFfor eachrow &columnx- 3x2x3When it looks like there is no common factor, take out a “1”. The sign still comes from the outer or inner term.+1+ 2x2- 3Answer: (2x2 - 3)(x + 1)