1. 5-4 Factoring Quadratic Expressions M11.A.1.2.1: Find the Greatest Common Factor and/or the Least Common Multiple for sets of monomials M11.D.2.1.5: Solve quadratic equations using factoring M11.D.2.2.2: Factor algebraic expressions, including differences of squares and trinomials

2. Objectives Finding Common Binomial Factors Factoring Special Expressions

3. Vocabulary Factoring is rewriting an expression as the product of its factors. The greatest common factor (GCF) of an expression is a common factor of the terms of the expression.

4. Factor each expression. a. 15x 2 + 25x + 100 15x 2 + 25x + 100 = 5(3x 2 ) + 5(5x) + 5(20) Factor out the GCF, 5 = 5(3x 2 + 5x + 20)Rewrite using the Distributive Property. b. 8m 2 + 4m 8m 2 + 4m = 4m(2m) + 4m(1) Factor out the GCF, 4m = 4m(2m + 1)Rewrite using the Distributive Property. Finding Common Factors

5. Factor x 2 + 10x + 24. Step 2:Rewrite the term bx using the factors you found. Group the remaining terms and find the common factors for each group. Step 1:Find factors with product ac and sum b. Factors of 24 Sum of factors 1, 24 25 2, 12 14 3, 8 11 6, 4 10 Since ac = 24 and b = 10, find positive factors with product 24 and sum 11. }} x 2 + 10x + 24 x 2 + 4x + 6x + 24Rewrite bx : 10x = 4x + 6x. x(x + 4) + 6(x + 4)Find common factors. Factoring when ac > 0 and b > 0

6. (continued) Step 3:Rewrite the expression as a product of two binominals. (x + 6)(x + 4)Rewrite using the Distributive Property. x(x + 4) + 6(x + 4) Check: (x + 6)(x + 4) = x 2 + 4x + 6x + 24 = x 2 + 10x + 24 Continued

7. Factor x 2 – 14x + 33. Step 1:Find factors with product ac and sum b. Step 2:Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as a product of two binomials. Factors of 33 Sum of factors –1, –33 –34 –3, –11 –14 Since ac = 33 and b = –14, find negative factors with product 33 and sum b. }} x 2 - 14x + 33 x 2 – 3x – 11x + 33Rewrite bx. x(x – 3) – 11(x – 3)Find common factors. (x – 11)(x – 3)Rewrite using the Distributive Property. Factoring when ac > 0 and b < 0

8. Factor x 2 + 3x –28. Step 1:Find factors with product ac and sum b. Step 2:Since a = 1, you can write binomials using the factors you found. x 2 + 3x – 28 (x – 4)(x + 7)Use the factors you found. Factors of –28 Sum of factors 1, –28 –27 –1, 28 27 2, –14 –12 –2, 14 12 4, –7 –3 –4, 7 3 Since ac = –28 and b = 3, find factors 2 with product –28 and sum 3. Factoring When ac < 0

9. Factor 6x 2 – 31x + 35. Step 1:Find factors with product ac and sum b. Step 2:Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as the product of two binomials. 2x(3x – 5) – 7(3x – 5)Find common factors. (2x – 7)(3x – 5)Rewrite using the Distributive Property. 6x 2 – 31x + 35 6x 2 – 10x – 21x + 35Rewrite bx. }} Factors of 210 Sum of factors -1, –210 –211 –2, –105 –107 –3, –70 –73 –5, –42 –47 –10, –21 –31 Since ac = 210 and b = –31, find negative factors with product 210 and sum –31. Factoring When a ≠ 1 and ac > 0

10. Factor 6x 2 + 11x – 35. Step 1:Find factors with product ac and sum b. Step 2:Rewrite the term bx using the factors you found. Then find common factors and rewrite the expression as the product of two binomials. 2x(3x – 5) + 7(3x – 5)Find common factors. (2x + 7)(3x – 5)Rewrite using the Distributive Property. 6x 2 + 11x - 35 6x 2 – 10x + 21x – 35Rewrite bx. Factors of –210 Sum of factors 1, –210 –209 –1, 210 209 2, –105 –103 –2, 105 103 3, –70 –67 Since ac = -210 and b = 11, find factors with product –210 and sum 11. Factors of –210 Sum of factors –3, 70 67 5, –42 –37 –5, 42 37 10, –21 –11 –10, 21 11 Factoring When a ≠ 1 and ac < 0

11. Vocabulary A perfect square trinomial is the product you obtain when you square a binomial. a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a – b)²

12. Factor 100x 2 + 180x + 81. 100x 2 + 180x + 81 = (10x) 2 + 180 + (9) 2 Rewrite the first and third terms as squares. = (10x) 2 + 180 + (9) 2 Rewrite the middle term to verify the perfect square trinomial pattern. = (10x + 9) 2 a 2 + 2ab + b 2 = (a + b) 2 Factoring a Perfect Square Trinomial

13. Homework Pg 259 #2-42 EOE