Bell Ringer 2/20/15 Completely Factor & Check your answer. 1.Factor: 2x x Factor: y 2 + 4y Factor: 75x 2 – 12

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. TypeNumber of Terms 1. GCF 2 or more 2. Grouping 4 3. Trinomials 3

First terms: Outer terms: Inner terms: Last terms: Combine like terms. y 2 + 4y + 4 y2y2 +2y +4 Review: Multiply (y + 2) 2 (y + 2)(y + 2) Check this out…whaaat!! (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Using the formula, (y + 2) 2 = (y) 2 + 2(y)(2) + (2) 2 (y + 2) 2 = y 2 + 4y + 4 Which one is quicker?

1) Factor x 2 + 6x + 9 Does this fit the form of our perfect square trinomial? 1)Is the first term a perfect square? Yes, a = x 2)Is the last term a perfect square? Yes, b = 3 3)Is the middle term twice the product of the a and b? Yes, 2ab = 2(x)(3) = 6x Perfect Square Trinomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Since all three are true, write your answer! (x + 3) 2 You can still factor the other way but this is quicker!

Bell work 1/15/14 Factor using any method 1.16xy y 2 z + 40y 2 2.6y y – 5 3.2r r + 18

2) Factor y 2 – 16y + 64 Does this fit the form of our perfect square trinomial? 1)Is the first term a perfect square? Yes, a = y 2)Is the last term a perfect square? Yes, b = 8 3)Is the middle term twice the product of the a and b? Yes, 2ab = 2(y)(8) = 16y Perfect Square Trinomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Since all three are true, write your answer! (y – 8) 2

Factor m 2 – 12m (m – 6)(m + 6) 2.(m – 6) 2 3.(m + 6) 2 4.(m – 18) 2

3) Factor 4p 2 + 4p + 1 Does this fit the form of our perfect square trinomial? 1)Is the first term a perfect square? Yes, a = 2p 2)Is the last term a perfect square? Yes, b = 1 3)Is the middle term twice the product of the a and b? Yes, 2ab = 2(2p)(1) = 4p Perfect Square Trinomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Since all three are true, write your answer! (2p + 1) 2

Does this fit the form of our perfect square trinomial? 1)Is the first term a perfect square? Yes, a = 5x 2)Is the last term a perfect square? Yes, b = 11y 3)Is the middle term twice the product of the a and b? Yes, 2ab = 2(5x)(11y) = 110xy 4) Factor 25x 2 – 110xy + 121y 2 Perfect Square Trinomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Since all three are true, write your answer! (5x – 11y) 2

Factor 9k k (3k + 2) 2 2.(3k – 2) 2 3.(3k + 2)(3k – 2) 4.I’ve got no clue…I’m lost!

Factor 2r r prime 2.2(r 2 + 6r + 9) 3.2(r – 3) 2 4.2(r + 3) 2 5.2(r – 3)(r + 3) Don’t forget to factor the GCF first!

Bell work 1/17/14 Factor the following m x x 2

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 or more 2. Grouping4 3. Trinomials3 4. Difference of Squares 2

Determine the pattern … = 1 2 = 2 2 = 3 2 = 4 2 = 5 2 = 6 2 These are perfect squares! You should be able to list the first 15 perfect squares … 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: Outer terms: Inner terms: Last terms: Combine like terms. x 2 – 4 x-2 x +2 x 2 +2x -2x -4 This is called the difference of squares. x2x2 +2x -2x -4 Notice the middle terms eliminate each other!

Difference of Squares a 2 - b 2 = (a - b)(a + b) or a 2 - b 2 = (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 x When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1 st term a perfect square? 2 nd term a perfect square? Subtraction? Write your answer! No Yes x 2 – 25 Yes ( )( )5xx+5 -

2. Factor 16x When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1 st term a perfect square? 2 nd term a perfect square? Subtraction? Write your answer! No Yes 16x 2 – 9 Yes (4x )(4x )3+3 -

When factoring, use your factoring table. Do you have a GCF? Are the Difference of Squares steps true? Two terms? 1 st term a perfect square? 2 nd term a perfect square? Subtraction? Write your answer! (9a )(9a )7b Factor 81a 2 – 49b 2 No Yes 81a 2 – 49b 2 Yes

Factor x 2 – y 2 1.(x + y)(x + y) 2.(x – y)(x + y) 3.(x + y)(x – y) 4.(x – y)(x – y) Remember, the order doesn’t matter!

When factoring, use your factoring table. Do you have a GCF? 3(25x 2 – 4) Are the Difference of Squares steps true? Two terms? 1 st term a perfect square? 2 nd term a perfect square? Subtraction? Write your answer! 3(5x )(5x ) Factor 75x 2 – 12 Yes! GCF = 3 Yes 3(25x 2 – 4) Yes

Factor 18c 2 + 8d 2 1.prime 2.2(9c 2 + 4d 2 ) 3.2(3c – 2d)(3c + 2d) 4.2(3c + 2d)(3c + 2d) You cannot factor using difference of squares because there is no subtraction!

Factor m 2 Rewrite the problem as 4m 2 – 64 so the subtraction is in the middle! 1.prime 2.(2m – 8)(2m + 8) 3.4(-16 + m 2 ) 4.4(m – 4)(m + 4)

Objective The student will be able to: use the zero product property to solve equations

Zero Product Property If a b = 0 then a=0, b=0, or both a and b equal 0.

1.Set the equation equal to 0. 2.Factor the equation. 3.Set each part equal to 0 and solve. 4.Check your answer on the calculator if available. 4 steps for solving a quadratic equation Set = 0 Factor Split/Solve Check

Using the Zero Product Property, you know that either x + 3 = 0 or x - 5 = 0 Solve each equation. x = -3 or x = 5 {-3, 5} 1. Solve (x + 3)(x - 5) = 0

2. Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = -4 or a = -7 a = -2 or a = -7 {-2, -7}

3. Solve (3t + 5)(t - 3) = 0 3t + 5 = 0 or t - 3 = 0 3t = -5 or t = 3 t = -5/3 or t = 3 {-5/3, 3}

Solve (y – 3)(2y + 6) = 0 1.{-3, 3} 2.{-3, 6} 3.{3, 6} 4.{3, -6}

4. Solve x x = 0 GCF = x x(x - 11) = 0 x = 0 or x - 11 = 0 x = 0 or x = 11 {0, 11} Set = 0 Factor Split/Solve Check

5. Solve. -24a +144 = -a 2 Put it in descending order. a a = 0 (a - 12) 2 = 0 a - 12 = 0 a = 12 {12} Set = 0 Factor Split/Solve Check

6. Solve 4m = 20m 4m m + 25 = 0 (2m - 5) 2 = 0 2m - 5 = 0 2m = 5 m = Set = 0 Factor Split/Solve Check

7. Solve x 3 + 2x 2 = 15x Set = 0 Factor Split/Solve Check x 3 + 2x x = 0 x(x 2 + 2x - 15) = 0 x(x + 5)(x - 3) = 0 x = 0 or x + 5 = 0 or x - 3 = 0 {0, -5, 3}

Solve a 2 – 3a = 40 1.{-8, 5} 2.{-5, 8} 3.{-8, -5} 4.{5, 8}

Solve 4r 3 – 16r = 0 1.{-16, 4} 2.{-4, 16} 3.{0, 2} 4.{0, 4} 5.{-2, 0, 2} The degree will tell you how many answers you have!

Bell Work 1/21/14 Grab sheet off green table Solve for Zero Property 1.4r 3 – 16r = 0 2.x 3 + 2x 2 = 15x

Maria told this puzzle to her friends. “The product of four times my age and 45 less than three times my age is zero. How old am I?” Find Maria’s age. Let m = Maria’s age. 4m(3m - 45) = 0 4m = 0 or 3m - 45 = 0 m = 0 or 3m = 45 m = 0 or m = 15 0 is not reasonable so Maria is 15 years old !!