Objectives The student will be able to: Factor using the

Objectives The student will be able to: Factor using the greatest common factor (GCF) and Factor by Grouping

Review: What is the GCF of 2 25a and 15a? 5a Let’s go one step further 1) FACTOR 25a2 15a. Find the GCF and divide each term by the GCF 25a2 15a 5a( 5a 3 ) 25a 2 5a 15a 5a Check your answer by distributing.

2) Factor 18x2 - 12x3. Find the GCF 6x2 Divide each term by the GCF 18x2 - 12x3 6x2( 3 - 2x ) 18 x 2 6 x2 12 x 3 6x2 Check your answer by distributing.

3) Factor 28a2b 56abc2. GCF 28ab Divide each term by the GCF 28a2b 56abc2 28ab ( a 2c2) 28a 2b 28ab 56abc 2 28ab Check your answer by distributing. 28ab(a 2c2)

Factor 20x2 - 24xy 1. 2. 3. 4. x(20 – 24y) 2x(10x – 12y) 4(5x2 – 6xy) 4x(5x – 6y)

2 2 2 5) Factor 28a 21b - 35b c GCF 7 Divide each term by the GCF 28a2 21b - 35b2c2 7 ( 4a2 3b - 5b2c2) 28a 2 7 21b 7 35b 2 c 2 7 Check your answer by distributing. 7(4a2 3b – 5b2c2)

Factor 16xy2 - 24y2z 40y2 1. 2. 3. 4. 2y2(8x – 12z 20) 4y2(4x – 6z 10) 8y2(2x - 3z 5) 8xy2z(2 – 3 5)

Factor by Grouping When polynomials contain four terms, it is sometimes easier to group like terms in order to factor. Your goal is to create a common factor. You can also move terms around in the polynomial to create a common factor. Practice makes you better in recognizing common factors.

Factor by Grouping Example 1: FACTOR: 3xy - 21y 5x – 35 Factor the first two terms: 3xy - 21y 3y (x – 7) Factor the last two terms: 5x - 35 5 (x – 7) The green parentheses are the same so it’s the common factor Now you have a common factor (x - 7) (3y 5)

Factor by Grouping Example 2: FACTOR: 6mx – 4m 3rx – 2r Factor the first two terms: 6mx – 4m 2m (3x - 2) Factor the last two terms: 3rx – 2r r (3x - 2) The green parentheses are the same so it’s the common factor Now you have a common factor (3x - 2) (2m r)

Factor by Grouping Example 3: FACTOR: 15x – 3xy 4y –20 Factor the first two terms: 15x – 3xy 3x (5 – y) Factor the last two terms: 4y –20 4 (y – 5) The green parentheses are opposites so change the sign on the 4 - 4 (-y 5) or – 4 (5 - y) Now you have a common factor (5 – y) (3x – 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. Solve (x 3)(x - 5) 0 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}

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

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. 2. 3. 4. {-3, 3} {-3, 6} {3, 6} {3, -6}

4 steps for solving a quadratic equation 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. Set 0 Factor Split/Solve Check

4. Solve x2 - 11x 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 -a2 Put it in descending order. a2 - 24a 144 0 (a - 12)2 0 a - 12 0 a 12 {12} Set 0 Factor Split/Solve Check

6. Solve 4m2 25 20m 4m2 - 20m 25 0 (2m - 5)2 0 2m - 5 0 2m 5 5 m 2 5 or 2.5 2 Set 0 Factor Split/Solve Check

7. Solve x3 2x2 15x x3 2x2 - 15x 0 Set 0 2 x(x 2x - 15) 0 Factor Split/Solve x(x 5)(x - 3) 0 Check x 0 or x 5 0 or x - 3 0 {0, -5, 3}

Solve a2 – 3a 40 1. 2. 3. 4. {-8, 5} {-5, 8} {-8, -5} {5, 8}

Solve 4r3 – 16r 0 1. 2. 3. 4. 5. {-16, 4} {-4, 16} {0, 2} {0, 4} {-2, 0, 2} The degree will tell you how many answers you have!

Find two consecutive integers whose product is 240. Let n 1st integer. Let n 1 2nd integer. n(n 1) 240 n2 n 240 n2 n – 240 0 (n – 15)(n 16) 0 Set 0 Factor Split/Solve Check

(n – 15)(n 16) 0 n – 15 0 or n 16 0 n 15 or n -16 The consecutive integers are 15, 16 or -16, -15.