Warm Up

Factor By Grouping

Goal We know how to write a general quadratic in vertex form (complete the square), but now we want to write a general quadratic in factored form.

When to use which method ResultMethod General to VertexCompleting the Square General to FactoredBy Grouping

First terms: Outer terms: Inner terms: Last terms: Combine like terms. y 2 + 6y + 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.

Steps 1. Factor out the GCF 2. Set up a MAMA table

Example 1 Factor y 2 + 6y + 8 Any GCF? No

Factor y 2 + 6y + 8 Create your MAMA table. MultiplyAdd 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

1) Factor y 2 + 6y + 8 Place the factors in the table. +1, +8 -1, -8 +2, +4 -2, -4 MultiplyAdd 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!

1) Factor y 2 + 6y , +4 MultiplyAdd , YES!! Hang with me now! Replace the middle number of the trinomial with our working numbers from the MAMA table y 2 + 6y + 8 y 2 + 2y + 4y + 8 Now, group the first two terms and the last two terms.

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? Factor out the GCF’s. Write them in their own group.

Example 2 Factor x 2 – 2x – 63

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

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! (x + 7)(x – 9)

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.

Example 3 5x x + 14

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

(5x 2 – 7x) (– 10x + 14) Factor out the GCF x(5x – 7) -2(5x – 7) The parentheses are the same! (x – 2)(5x – 7) These will continue to get easier the more you do them.

You try!Factor x 2 + 3x (x + 2)(x + 1) 2.(x – 2)(x + 1) 3.(x + 2)(x – 1) 4.(x – 2)(x – 1)

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

You try!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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