2. ALGEBRA 1 Lesson 8-7 Warm-Up

3. ALGEBRA 1 “Factoring Special Cases” (8-7) What is a “perfect square trinomial”? How do you factor a “perfect square trinomial”? Perfect Square Trinomial: a trinomial in the form of: a 2 + 2ab + b 2 or a 2 - 2ab + b 2 In other words, the middle term is twice the product of one a and one b (actor of the third term). Rule: Perfect Square Trinomials: : a 2 + 2ab + b 2 = (a + b)(a + b) = (a + b) 2 a 2 - 2ab + b 2 = (a - b)(a - b) = (a - b) 2 Examples: x 2 + 10x + 25 = (x + 5)(x + 5) = (x + 5) 2 x 2 - 10x + 25 = (x - 5)(x - 5) = (x - 5) 2 S

4. ALGEBRA 1 “Factoring Special Cases” (8-7) How can you recognize a perfect square trinomial? Tip: To recognize a perfect square trinomial: 1.The first and last terms can both be written as the product of two identical factors. 2.The middle term is twice the product of a factor of the first terms and a factor of the third term. Example: 4x 2 + 12x + 94x 2 + 20x + 9 2x2x 33 2x2x 33 2(2x 3) = 12x  2(2x 3) ≠ 12x Area Model: S 2x + 3

5. ALGEBRA 1 Factor m 2 – 6m + 9. m 2 – 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. Factoring Special Cases LESSON 8-7 Additional Examples

6. ALGEBRA 1 The area of a square is (16h 2 + 40h + 25) in. 2 Find the length of a side. 16h 2 + 40h + 25 = (4h) 2 + 40h + 5 2 Write 16h 2 as (4h) 2 and 25 as 5 2. = (4h + 5) 2 Write the factors as the square of a binomial. The side of the square has a length of (4h + 5) in. = (4h) 2 + 2(4h)(5) + 5 2 Does the middle term equal 2ab? 40h = 2(4h)(5) Factoring Special Cases LESSON 8-7 Additional Examples (4h + 5)

7. ALGEBRA 1 “Factoring Special Cases” (8-7) How do you factor the “difference of two squares”? Rule: Recall that the difference of two squares is a 2 - b 2. The factors of the difference of two squares is the product of the sum and difference of a and b. a 2 - b 2 = (a + b)(a - b) Examples: Factor x 2 - 64 x 2 - 64 = x 2 - 8 2 Rewrite 64 as 8 2 so the first and second terms are both squared (x + 8)(x - 8) Factor using the difference of two squares rule Check (using FOIL) : (x + 8)(x - 8) = x 2 + 8x - 8x - 8 2 = x 2 - 8 2 = x 2 - 64  S

8. ALGEBRA 1 Factor a 2 – 16. a 2 – 16 = a 2 – 4 2 Rewrite 16 as 4 2. = (a + 4)(a – 4) Factor. Check: Use FOIL to multiply. (a + 4)(a – 4) a 2 – 4a + 4a – 16 a 2 – 16 Factoring Special Cases LESSON 8-7 Additional Examples

9. ALGEBRA 1 Factor 9b 2 – 225. 9b 2 – 225 = (3b) 2 – 15 2 Rewrite 9b 2 as (3b) 2 and 225 as 15 2. = (3b + 15)(3b –15) Factor. Factoring Special Cases LESSON 8-7 Additional Examples

10. ALGEBRA 1 Factor 5x 2 – 80. 5x 2 – 80 = 5(x 2 – 16)Factor out the common factor of 5. = 5(x + 4)(x – 4)Factor (x 2 – 16). Check: Use FOIL to multiply the binomials. Then multiply by common factor. 5(x + 4)(x – 4) 5(x 2 – 16) 5x 2 – 80 Factoring Special Cases LESSON 8-7 Additional Examples = 5(x 2 – 4 2 )Turn the binomial into the difference of two squares.

11. ALGEBRA 1 Factor each expression. 1.y 2 – 18y + 812.9a 2 – 24a + 16 3.p 2 – 1694.36x 2 – 225 5.5m 2 – 456.2c 2 + 20c + 50 (y – 9) 2 (3a – 4) 2 (p + 13)(p – 13)9(2x + 5)(2x – 5) 5(m + 3)(m – 3)2(c + 5) 2 Factoring Special Cases LESSON 8-7 Lesson Quiz