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Factoring

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Objective The student will be able to: Factor trinomials with grouping and Trial & Error. MM1A2f

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First terms: Outer terms: Inner terms: Last terms: Combine like terms. y 2 + 6y + 8 y+2 y +4 y2y2 +4y +2y +8 y2y2 +4y +2y +8 Review: (y + 2)(y + 4) In this lesson, we will begin with y 2 + 6y + 8 as our problem and finish with (y + 2)(y + 4) as our answer.

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Here we go! 1) Factor y 2 + 6y + 8 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.

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1) Factor y 2 + 6y + 8 Create your MAMA table. MultiplyAdd +8 +6 Product of the first and last coefficients 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. M A

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1) Factor y 2 + 6y + 8 Place the factors in the table. +1, +8 -1, -8 +2, +4 -2, -4 MultiplyAdd +8 +6 Which has a sum of +6? +9, NO -9, NO +6, YES!! -6, NO We are going to use these numbers in the next step!

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1) Factor y 2 + 6y + 8 +2, +4 MultiplyAdd +8 +6 +6, YES!! Hang with me now! Create two new binomials with your variable and the values you found in your MAMA table: ( )( ) y y+ 2 + 4

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1) Factor. y 2 + 6y + 8 Put the first and last terms into the box as shown. What are the factors of y 2 ? y and y y2y2 + 8

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1) Factor. y 2 + 6y + 8 Place the factors outside the box as shown. y2y2 + 8 y y What are the factors of + 8? +1 and +8, -1 and -8 +2 and +4, -2 and -4

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The second box works. Write the numbers on the outside of box for your solution. 1) Factor. y 2 + 6y + 8 Which box has a sum of + 6y? y2y2 + 8 y y y2y2 y y+ 1+ 2 y + 8 y + 4 + y + 8y+ 4y + 2y

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1) Factor. y 2 + 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.

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2) Factor x 2 – 2x – 63 Create your MAMA table. MultiplyAdd -63 -2 Product of the first and last coefficients Middle coefficient -63, 1 -1, 63 -21, 3 -3, 21 -9, 7 -7, 9 -62 62 -18 18 -2 2 Signs need to be different since number is negative. M A

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1) Factor x 2 – 2x – 63 7, - 9 MultiplyAdd -63 -2 -2, YES!!

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x2x2 - 63 2) Factor. x 2 - 2x - 63 Put the first and last terms into the box as shown. What are the factors of x 2 ? x and x

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2) Factor. x 2 - 2x - 63 Place the factors outside the box as shown. x2x2 - 63 x x What are the factors of - 63? Remember the signs will be different!

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2) Factor. x 2 - 2x - 63 Use trial and error to find the correct combination! Do any of these combinations work? The second one has the wrong sign! x2x2 - 63 x x + 21 - 3 +21x -3x x2x2 - 63 x x- 7 + 9 -7x +9x

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2) Factor. x 2 - 2x - 63 Change the signs of the factors! Write your solution. (x + 7)(x - 9) x2x2 - 63 x x+ 7 - 9 +7x -9x

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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.

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Factor x 2 + 3x + 2 1.(x + 2)(x + 1) 2.(x – 2)(x + 1) 3.(x + 2)(x – 1) 4.(x – 2)(x – 1)

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Factoring Chart This chart will help you to determine which method of factoring to use. TypeNumber of Terms 1. GCF 2 or more 2. Diff. Of Squares 2 3. Trinomials 3

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We have two groups! (y 2 + 2y)(+4y + 8) If things are done right, the parentheses should be the same. Almost done! Find the GCF of each group and factor it out. y(y + 2) +4(y + 2) (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…) Factor out the GCF’s. Write them in their own group.

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Replace the middle term with our working numbers. x 2 – 2x – 63 x 2 – 9x + 7x – 63 Group the terms. (x 2 – 9x) (+ 7x – 63) Factor out the GCF x(x – 9) +7(x – 9) The parentheses are the same! Weeedoggie! (x + 7)(x – 9)

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2) Factor 5x 2 - 17x + 14 Create your MAMA table. MultiplyAdd +70 -17 Product of the first and last coefficients Middle coefficient -1, -70 -2, -35 -7, -10 -71 -37 -17 Signs need to be the same as the middle sign since the product is positive. M A

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(5x 2 – 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.

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Factor 2x 2 + 9x + 10 1.(2x + 10)(x + 1) 2.(2x + 5)(x + 2) 3.(2x + 2)(x + 5) 4.(2x + 1)(x + 10)

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Factor 6y 2 – 13y – 5 1.(6y 2 – 15y)(+2y – 5) 2.(2y – 1)(3y – 5) 3.(2y + 1)(3y – 5) 4.(2y – 5)(3y + 1)

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2) Factor 2x 2 - 14x + 12 MultiplyAdd +6 -7 Find the GCF! 2(x 2 – 7x + 6) Now do the MAMA table! -7 -5 Signs need to be the same as the middle sign since the product is positive. Replace the middle term. 2[x 2 – x – 6x + 6] Group the terms. -1, -6 -2, -3

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2[(x 2 – 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!!

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