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LESSON 7: FACTORING SPECIAL POLYNOMIALS Learning Outcome: Learn to investigate some special factoring patterns. With a partner, determine each product: (x + 1)² (x + 2)² (x + 3)² (x - 1)² (x – 2)² (x – 3)² (2x + 1)² (3x + 1)² (4x + 1)² (2x – 1)² (3x – 1)² (4x – 1)² What patterns do you see in the trinomials and their factors above? How could you use the patterns to factor these trinomials? 4x² + 20x + 25 9x² - 12x + 4 Write a strategy for factoring polynomials of this type. When given the situation of x 2 . The term “squaring” means multiplying the number by itself. 2 Ex. x 4 x 4 x 4 2 Ex. 3x 5 3x 5 3x 5 2 Ex. 2 General expansion of squared binomials or Perfect Square Trinomials: a b a 2 2ab b2 2 a b a 2 2ab b2 2 Factoring Perfect Square Trinomials: Patterns: a b a 2 2ab b 2 and 2 a b 2 a 2 2ab b 2 Factoring PST: 4 x 2 20 x 25 - recognize it first how recognize: 1 and last term are perfect squares st a 4x2 4x2 2x c 25 25 5 b 2ab 2 2 x 5 20 x It works, all the numbers match and add up correctly. Now to factor: 4 x 2 20 x 25 square root the 1st and last terms: a = 2x, c = 5 To write in form pay attention to the sign in front of the b value: (in this case +) 2 Write in form: 2 x 5 Ex. Factor : a. 25a 2 20a 4 b. 4a 2 4ab b2 Difference of Squares: Ex. x 1 x 1 x2 x x 1 x2 1 Ex. 3x 53x 5 9 x2 15x 15x 25 9 x2 25 Ex. 2 2 2 2 Notice that the two terms are subtracted and each term is squared (Difference of squares) General Rule: a b a b a2 b2 Factoring Difference of Squares: Pattern: a2 b2 a b a b Factoring DOS: 9 x 2 16 9x Ex. Factor each: a. 9 x 2 25 2 - recognize it first 16 - square root the 1st and last terms 3x 4 Write in form: (3x-4)(3x+4) b. 100 y 2 9 x 2 c. 16m2 4n2 d. 162𝑣 4 − 2𝑤 4 Assignment: pg. 194-195 #4-8, 10-13, 15, 18, 20