Factoring Special Cases ALGEBRA 1 LESSON 7-7 Factor m2 – 6m + 9. m2 – 6m + 9 = m • m – 6m + 3 • 3 Rewrite first and last terms. = m • m – 2(m • 3) + 3 • 3 Does the middle term equal 2ab? 6m = 2(m • 3) = (m – 3)2 Write the factors as the square of a binomial. 7-7

Factoring Special Cases ALGEBRA 1 LESSON 7-7 The area of a square is (16h2 + 40h + 25) in.2. Find the length of a side. 16h2 + 40h + 25 = (4h)2 + 40h + 52 Write 16h2 as (4h)2 and 25 as 52. = (4h)2 + 2(4h)(5) + 52 Does the middle term equal 2ab? 40h = 2(4h)(5) = (4h + 5)2 Write the factors as the square of a binomial. The side of the square has a length of (4h + 5) in. 7-7

Factoring Special Cases ALGEBRA 1 LESSON 7-7 Factor a2 – 16. a2 – 16 = a2 – 42 Rewrite 16 as 42. = (a + 4)(a – 4) Factor. Check: Use FOIL to multiply. (a + 4)(a – 4) a2 – 4a + 4a – 16 a2 – 16 7-7

Factoring Special Cases ALGEBRA 1 LESSON 7-7 Factor 9b2 – 25. 9b2 – 225 = (3b)2 – 52 Rewrite 9b2 as (3b)2 and 25 as 52. = (3b + 5)(3b – 5) Factor. 7-7

Factoring Special Cases ALGEBRA 1 LESSON 7-7 Factor 5x2 – 80. 5x2 – 80 = 5(x2 – 16) Factor out the GCF of 5. = 5(x + 4)(x – 4) Factor (x2 – 16). Check: Use FOIL to multiply the binomials. Then multiply by the GCF. 5(x + 4)(x – 4) 5(x2 – 16) 5x2 – 80 7-7

= (2x + 1)(3x2 – 2) Factor out (2x + 1). Factoring by Grouping ALGEBRA 1 LESSON 7-7 Factor 6x3 + 3x2 – 4x – 2. 6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the GCF from each group of two terms. = (2x + 1)(3x2 – 2) Factor out (2x + 1). = 6x3 – 4x + 3x2 – 2 Use FOIL. Check: 6x3 + 3x2 – 4x – 2 (2x + 1)(3x2 – 2) = 6x3 + 3x2 – 4x – 2 Write in standard form. 7-7

8t4 + 12t3 + 16t + 24 = 4(2t4 + 3t3 + 4t + 6) Factor out the GCF, 4. Factoring by Grouping ALGEBRA 1 LESSON 7-7 Factor 8t4 + 12t3 + 16t + 24. 8t4 + 12t3 + 16t + 24 = 4(2t4 + 3t3 + 4t + 6) Factor out the GCF, 4. = 4[t3(2t + 3) + 2(2t + 3)] Factor by grouping. = 4(2t + 3)(t3 + 2) Factor again. 7-7

Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3) Factor out the GCF, 2. Factoring by Grouping ALGEBRA 1 LESSON 7-7 Factor 24h2 + 10h – 6. Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3)   Factor out the GCF, 2. Step 2: 12 • –3 = –36 Find the product ac. Step 3: Factors Sum –2(18) = –36 –2 + 18 = 16 –3(12) = –36 –3 + 12 = 9 –4(9) = –36 –4 + 9 = 5 Find two factors of ac that have a sum b. Use mental math to determine a good place to start. Step 4:  12h2 – 4h + 9h – 3 Rewrite the trinomial. Step 5:   4h(3h – 1) + 3(3h – 1) Factor by grouping.   (4h + 3)(3h – 1) Factor again. 24h2 + 10h – 6 = 2(4h + 3)(3h – 1) Include the GCF in your final answer. 7-7

Factoring Special Cases ALGEBRA 1 LESSON 7-7 Factor each expression. 1. y2 – 18y + 81 2. 9a2 – 24a + 16 3. p2 – 169 4. 36x2 – 225 5. 5m2 – 45 6. 2c2 + 20c + 50 (y – 9)2 (3a – 4)2 (p + 13)(p – 13) (6x + 15)(6x – 15) 5(m + 3)(m – 3) 2(c + 5)2 7-7

Factor each expression. 1. 10p3 – 25p2 + 4p – 10 Factoring by Grouping ALGEBRA 1 LESSON 7-7 Factor each expression. 1. 10p3 – 25p2 + 4p – 10 2. 36x4 – 48x3 + 9x2 – 12x 3. 16a3 – 24a2 + 12a – 18 (5p2 + 2)(2p – 5) 3x(4x2 + 1)(3x – 4) 2(4a2 + 3)(2a – 3) 7-7