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Bell Work 1/10/14 1. Multiply using Box method (2a – 3b)(2a + 4b) 2. Complete #’s 47 and 49 from page 2 on HW pages. Objective The student will be able to: factor trinomials with grouping by AC (MAMA) method. Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2. Grouping 3. Trinomials 2 or more 4 3 Review: (y + 2)(y + 4) y2 +4y +2y +8 First terms: Outer terms: Inner terms: Last terms: Combine like terms. y2 + 6y + 8 y +2 y2 +2y +4 +4y +8 y In this lesson, we will begin with y2 + 6y + 8 as our problem and finish with (y + 2)(y + 4) as our answer. Here we go! 1) Factor y2 + 6y + 8 Use your factoring chart. Do we have a GCF? Nope! Now we will learn Trinomials! You will set up a table with the following information. Product of the first and last coefficients Middle coefficient The goal is to find two factors in the first column that add up to the middle term in the second column. We’ll work it out in the next few slides. 1) Factor 2 y M A + 6y + 8 Create your MAMA table. Product of the first and last coefficients Multiply +8 Add +6 Middle coefficient Here’s your task… What numbers multiply to +8 and add to +6? If you cannot figure it out right away, write the combinations. 1) Factor 2 y + 6y + 8 Place the factors in the table. Multiply +8 Which has a sum of +6? +1, +8 -1, -8 +2, +4 -2, -4 Add +6 +9, NO -9, NO +6, YES!! -6, NO We are going to use these numbers in the next step! 1) Factor y2 + 6y + 8 Multiply +8 Add +6 +2, +4 +6, YES!! Hang with me now! Replace the middle number of the trinomial with our working numbers from the MAMA table y2 + 6y + 8 y2 + 2y + 4y + 8 Now, group the first two terms and the last two terms. We have two groups! (y2 + 2y)(+4y + 8) Almost done! Find the GCF of each group and factor it out. If things are done right, the parentheses y(y + 2) +4(y + 2) should be the same. Factor out the GCF’s. Write them in their own group. (y + 4)(y + 2) Tadaaa! There’s your answer…(y + 4)(y + 2) You can check it by multiplying. Piece of cake, huh? There is a shortcut for some problems too! (I’m not showing you that yet…) M A 2) Factor x2 – 2x – 63 Create your MAMA table. Product of the first and last coefficients Signs need to be different since number is negative. Multiply -63 -63, 1 -1, 63 -21, 3 -3, 21 -9, 7 -7, 9 Add -2 -62 62 -18 18 -2 2 Middle coefficient Replace the middle term with our working numbers. x2 – 2x – 63 x2 – 9x + 7x – 63 Group the terms. (x2 – 9x) (+ 7x – 63) Factor out the GCF x(x – 9) +7(x – 9) The parentheses are the same! Weeedoggie! (x + 7)(x – 9) Bell Work 1/13/14 1. Factor by grouping 16k3 - 4k2p2 - 28kp +7p3 2. Factor using AC method 2x2 + 9x + 10 Here are some hints to help you choose your factors in the MAMA table. 1) When the last term is positive, the factors will have the same sign as the middle term. 2) When the last term is negative, the factors will have different signs. M A 2) Factor 5x2 - 17x + 14 Create your MAMA table. Product of the first and last coefficients Signs need to be the same as the middle sign since the product is positive. Multiply +70 -1, -70 -2, -35 -7, -10 Add -17 -71 -37 -17 Replace the middle term. 5x2 – 7x – 10x + 14 Group the terms. Middle coefficient (5x2 – 7x) (– 10x + 14) Factor out the GCF x(5x – 7) -2(5x – 7) The parentheses are the same! Weeedoggie! (x – 2)(5x – 7) Hopefully, these will continue to get easier the more you do them. Factor 1. 2. 3. 4. (x + 2)(x + 1) (x – 2)(x + 1) (x + 2)(x – 1) (x – 2)(x – 1) 2 x + 3x + 2 Factor 1. 2. 3. 4. 2 6y (6y2 – 15y)(+2y – 5) (2y – 1)(3y – 5) (2y + 1)(3y – 5) (2y – 5)(3y + 1) – 13y – 5 2) Factor 2x2 - 14x + 12 Find the GCF! 2(x2 – 7x + 6) Now do the MAMA table! Signs need to be the same as the middle sign since the product is positive. Multiply +6 Add -7 -1, -6 -7 -2, -3 -5 Replace the middle term. 2[x2 – x – 6x + 6] Group the terms. 2[(x2 – x)(– 6x + 6)] Factor out the GCF 2[x(x – 1) -6(x – 1)] The parentheses are the same! Weeedoggie! 2(x – 6)(x – 1) Don’t forget to follow your factoring chart when doing these problems. Always look for a GCF first!! Objective The student will be able to: factor quadratic trinomials. Trial and Error Method Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3 Review: (y + 2)(y + 4) Multiply using FOIL or using the Box Method. Box Method: y + 4 y y2 +4y + 2 +2y +8 Combine like terms. 2 FOIL: y + 4y + 2y + 8 y2 + 6y + 8 1) Factor. 2 y + 6y + 8 Put the first and last terms into the box as shown. y2 +8 What are the factors of y2? y and y Objective The student will be able to: factor perfect square trinomials. Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2. Grouping 3. Trinomials 2 or more 4 3 Review: Multiply (y + 2)2 (y + 2)(y + 2) Do you remember these? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 y2 Using the formula, +2y (y + 2)2 = (y)2 + 2(y)(2) + (2)2 2 = y2 + 4y + 4 (y + 2) +2y +4 Which one is quicker? First terms: Outer terms: Inner terms: Last terms: Combine like terms. y2 + 4y + 4 1) Factor x2 + 6x + 9 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 1) Is the first term a perfect square? Yes, a = x Since all three are true, 2) Is the last term a perfect write your answer! square? (x + 3)2 Yes, b = 3 3) Is the middle term twice the You can still product of the a and b? factor the other way but this is quicker! Yes, 2ab = 2(x)(3) = 6x Bell work 1/15/14 Factor using any method 1. 16xy2 - 24y2z + 40y2 2. 6y2 - 13y – 5 3. 2r2 + 12r + 18 2) Factor y2 – 16y + 64 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 1) Is the first term a perfect square? Yes, a = y Since all three are true, 2) Is the last term a perfect write your answer! square? (y – 8)2 Yes, b = 8 3) Is the middle term twice the product of the a and b? Yes, 2ab = 2(y)(8) = 16y Factor m2 – 12m + 36 1. 2. 3. 4. (m – 6)(m + 6) (m – 6)2 (m + 6)2 (m – 18)2 3) Factor 4p2 + 4p + 1 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 1) Is the first term a perfect square? Yes, a = 2p Since all three are true, 2) Is the last term a perfect write your answer! square? (2p + 1)2 Yes, b = 1 3) Is the middle term twice the product of the a and b? Yes, 2ab = 2(2p)(1) = 4p 4) Factor 25x2 – 110xy + 121y2 Does this fit the form of our Perfect Square Trinomials perfect square trinomial? (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 – 2ab + b2 1) Is the first term a perfect square? Yes, a = 5x Since all three are true, 2) Is the last term a perfect write your answer! square? (5x – 11y)2 Yes, b = 11y 3) Is the middle term twice the product of the a and b? Yes, 2ab = 2(5x)(11y) = 110xy Factor 9k2 + 12k + 4 1. 2. 3. 4. (3k + 2)2 (3k – 2)2 (3k + 2)(3k – 2) I’ve got no clue…I’m lost! Bell work 1/16/14 Factor: 24x2 + 22x - 10 Bell work 1/17/14 Factor the following. 1. 75x2 – 12 2. -64 + 4m2 3. 50x - 60 + 10x2 Objective The student will be able to: factor using difference of squares. Factoring Chart This chart will help you to determine which method of factoring to use. Type Number of Terms 1. GCF 2. Grouping 3. Trinomials -AC Method, Trial & Error, Perfect Squares 4. Difference of Squares 2 or more 4 3 2 Determine the pattern 1 4 9 16 25 36 … = 12 = 22 = 32 = 42 = 52 = 62 These are perfect squares! You should be able to list the first 15 perfect squares in 30 seconds… Perfect squares 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 Review: Multiply (x – 2)(x + 2) First terms: x2 Outer terms: +2x Inner terms: -2x Last terms: -4 Combine like terms. x2 – 4 Notice the middle terms eliminate each other! x -2 x2 -2x +2 +2x -4 x This is called the difference of squares. Difference of Squares 2 2 a - b = (a - b)(a + b) or 2 2 a - b = (a + b)(a - b) The order does not matter!! 4 Steps for factoring Difference of Squares 1. Are there only 2 terms? 2. Is the first term a perfect square? 3. Is the last term a perfect square? 4. Is there subtraction (difference) in the problem? If all of these are true, you can factor using this method!!! 1. Factor x2 - 25 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? x2 – 25 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes ( x + 5 )(x - 5 ) Write your answer! 2. Factor 16x2 - 9 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? 16x2 – 9 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes (4x + 3 )(4x - 3 ) Write your answer! 3. Factor 81a2 – 49b2 When factoring, use your factoring table. Do you have a GCF? No Are the Difference of Squares steps true? 81a2 – 49b2 Two terms? Yes 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes (9a + 7b)(9a - 7b) Write your answer! Factor 1. 2. 3. 4. 2 x (x + y)(x + y) (x – y)(x + y) (x + y)(x – y) (x – y)(x – y) Remember, the order doesn’t matter! – 2 y 4. Factor 2 75x – 12 When factoring, use your factoring table. Do you have a GCF? Yes! GCF = 3 3(25x2 – 4) Are the Difference of Squares steps true? Two terms? Yes 3(25x2 – 4) 1st term a perfect square? Yes 2nd term a perfect square? Yes Subtraction? Yes 3(5x + 2 )(5x - 2 ) Write your answer! Factor 1. 2. 3. 4. 2 18c prime 2(9c2 + 4d2) 2(3c – 2d)(3c + 2d) 2(3c + 2d)(3c + 2d) You cannot factor using difference of squares because there is no subtraction! + 2 8d Factor -64 + 1. 2. 3. 4. prime (2m – 8)(2m + 8) 4(-16 + m2) 4(m – 4)(m + 4) Rewrite the problem as 4m2 – 64 so the subtraction is in the middle! 2 4m 1) Factor. 2 y + 6y + 8 Place the factors outside the box as shown. y y y2 +8 What are the factors of + 8? +1 and +8, -1 and -8 +2 and +4, -2 and -4 1) Factor. 2 y + 6y + 8 Which box has a sum of + 6y? y +1 y +2 y 2 y + 8 + 8y +y +8 y 2 y + 4 + 4y + 2y +8 The second box works. Write the numbers on the outside of box for your solution. 1) Factor. 2 y + 6y + 8 (y + 2)(y + 4) Here are some hints to help you choose your factors. 1) When the last term is positive, the factors will have the same sign as the middle term. 2) When the last term is negative, the factors will have different signs. 2) Factor. 2 x - 2x - 63 Put the first and last terms into the box as shown. x2 - 63 What are the factors of x2? x and x 2) Factor. 2 x - 2x - 63 Place the factors outside the box as shown. x x 2 x - 63 What are the factors of - 63? Remember the signs will be different! 2) Factor. x2 - 2x - 63 Use trial and error to find the correct combination! x -3 x -7 x x2 -3x + 21 +21x - 63 x x2 + 9 +9x -7x - 63 Do any of these combinations work? The second one has the wrong sign! 2) Factor. x2 - 2x - 63 Change the signs of the factors! x +7 x -9 2 x +7x -9x - 63 Write your solution. (x + 7)(x - 9) Bell Work 1/14/14 On new work sheet on green table • Complete #’s 10,12 by AC method • Complete # 14 using Trial and Error method 3) Factor. 2 5x - 17x + 14 Put the first and last terms into the box as shown. 5x2 + 14 What are the factors of 5x2? 5x and x 3) Factor. 5x2 - 17x + 14 5x x 5x2 + 14 What are the factors of + 14? Since the last term is positive, the signs of the factors are the same! Since the middle term is negative, the factors must be negative! 3) Factor. 5x2 - 17x + 14 When the coefficient is not 1, you must try both combinations! 5x -2 5x -7 x -7 5x2 - 2x x -35x + 14 - 2 5x2 - 7x -10x + 14 Do any of these combinations work? The second one! Write your answer. 3) Factor. 5x2 - 17x + 14 (5x - 7)(x - 2) It is not the easiest of things to do, but the more problems you do, the easier it gets! Trust me! 2 2x 4) Factor + 9x + 10 (x + 2)(2x + 5) 5) Factor. 2 6y - 13y - 5 (2y - 5)(3y + 1) 6) 12x2 + 11x - 5 (4x + 5)(3x - 1) 7) 5x - 6 + x2 2 x + 5x - 6 (x - 1)(x + 6)