Presentation on theme: "SLIDE SHOW INSTRUCTIONS"— Presentation transcript:
1 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.(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)
2 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
3 Factoring Trinomials: 1. Split the middle term2. Factor using the box
4 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 4-terms can be put into the box and factored immediately.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”.
5 Why do some polynomials have 3 terms and others have 4? When the FOIL multiplication methodis used to multiply 2 binomials,for example: (x + 3)(4x - 2) or (a + b)(c - d)Each answer will have four terms (First, Outer, Inner, Last)If the Outer and Inner terms can be combined,the result will be a Trinomial (3 terms).(x + 3)(4x - 2)= x2 - 2x + 12x - 6 = x2 + 10x - 6If they cannot be combined, it keeps all 4 terms.(a + b)(c - d) = ac - ad + bc - bdAn earlier lesson covered the process for factoring 4-terms, this tutorial covers factoring 3-term polynomials.
6 II. Factoring a Trinomial of the form: ax2 + bx + c 3 terms: 2x2 + 9x + 10First(a)Middle(b)Last(c)First, we will split the middle term into the original outer andinner terms. This gives us a four term polynomial which wecan factor using the Greenebox method.This is the other application for the 2-dimensional factoring method. The first part of this procedure is borrowed from a factoring method called the ac-method. The ac-method, splits the middle term into the original outer and inner terms then uses grouping. Since the box is an alternative to grouping, we use the ac-method to split the middle term, then use the box instead of grouping to factor.There are two examples here. One with a positive last term and one with a negative last term.
7 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)Step 5: Choose the correct signs.FACT: We add numbers when they have the same signs. (step 3).i.e. The factors are both positive, + 4x and + 5x OR both negative, -4x and -5xThis 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.
8 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
9 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
10 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
11 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
12 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- 10
13 SPECIAL CASE!!!There are some polynomials that have a factor thatmust be taken out before using the box.Take a polynomial such as:4x2 + 10x - 6This 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.
14 Step 4: Choose the pair of factors that equal + 10x The way this problem should NOT be workedThis problem has a GCF of “2”. We’ll pretend we didn’t see it and work the problem normally,4x2 + 10x - 6-Step 3:Subtract4 * 6 = 24Step 1: Multiplythe first*last1 * 242 * 123 * 84 * 6= 23= 10= 5= 2Step 2: Find the factors of 24Step 4: Choose the pair of factors that equal + 10x(the middle term)Step 5: Choose the correct signs:-2x and +12x = +10xor 2x and -12x = -10x
15 4x2 - 2x - 6 2x + 6 4x + 12x - 2 (2x + 6)(4x - 2)= 8x2 + 20x - 12X The way this problem should NOT be worked4x2 + 12x - 2x - 6Put the four factors into the box and factor it2x+ 6If we checkthe answer using FOIL, weDO NOT getthe original problem!!!4x2 + 10x - 6The box“SEES”the “2”in bothdimensionsand pulls itout twice!!!4x4x2+ 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!!!
16 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
17 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