Warm-up #23 (Monday, 10/26) Simplify (x+2)(3x – 4) 2. Simplify (x+2)( x 2 −3x−1) 3. Simplify 3 2 −1 2 2

Homework (Monday, 10/26) Factoring packet_pg 1 Lesson 2.06 worksheet

Lesson 2.10 Factoring a Difference of Squares

Factoring 3x+3 = 3(x+1) 3xy – 3x= 3x(y – 1)

Ex: Factor x(a + b) – 2(a + b) Always ask first if there is common factor the terms share . . . x(a + b) – 2(a + b) Each term has factor (a + b)  x(a + b) – 2(a + b) = (a + b)( – ) x 2 (a + b) (a + b)  x(a + b) – 2(a + b) = (a + b)(x – 2)

Ex: Factor a(x – 2) + 2(2 – x) As with the previous example, is there a common factor among the terms? Well, kind of . . . x – 2 is close to 2 - x . . . Hum . . . Recall: (-1)(x – 2) = - x + 2 = 2 – x  a(x – 2) + 2(2 – x) = a(x – 2) + 2((-1)(x – 2)) = a(x – 2) + (– 2)(x – 2) = a(x – 2) – 2(x – 2)  a(x – 2) – 2(x – 2) = (x – 2)( – ) a 2 (x – 2) (x – 2)

Ex: Factor b(a – 7) – 3(7 – a) Common factor among the terms? Well, kind of . . . a – 7 is close to 7 - a Recall: (-1)(a – 7) = - a + 7 = 7 – a  b(a – 7) – 3(7 – a) = b(a – 7) – 3((-1)(a – 7)) = b(a – 7) + 3(a – 7) = b(a – 7) +3(a – 7)  b(a – 7) + 3(a – 7) = (a – 7)( + ) b 3 (a – 7) (a – 7)

The difference of squares For any numbers “a” and “b”, the following is an identity. a2 – b2 = (a – b)(a + b) Note: you can replace “a” and “b” by any number, variable, or expression LIST ALL OF THE PERFECT SQUARES:

Ex: Factor x2 – 4 Notice the terms are both perfect squares and we have a difference  difference of squares x2 = (x)2 4 = (2)2  x2 – 4 = (x)2 – (2)2 = (x – 2)(x + 2) a2 – b2 = (a – b)(a + b) factors as

Example 1: 𝑥 2 −49 Example 2: 𝑥 2 −125

Ex: Factor 9p2 – 16 Notice the terms are both perfect squares and we have a difference  difference of squares 9p2 = (3p)2 16 = (4)2  9a2 – 16 = (3p)2 – (4)2 = (3p – 4)(3p + 4) a2 – b2 = (a – b)(a + b) factors as

Example 1: 25𝑥 2 −16 Example 2: 15𝑥 2 −100

Ex: Factor y6 – 25 Notice the terms are both perfect squares and we have a difference  difference of squares y6 = (y3)2 25 = (5)2  y6 – 25 = (y3)2 – (5)2 = (y3 – 5)(y3 + 5) a2 – b2 = (a – b)(a + b) factors as

Example 1: 𝑚 8 −36 Example 2: 𝑥 6 −64

Ex: Factor 81 – x2y2 Notice the terms are both perfect squares and we have a difference  difference of squares 81 = (9)2 x2y2 = (xy)2  81 – x2y2 = (9)2 – (xy)2 = (9 – xy)(9 + xy) a2 – b2 = (a – b)(a + b) factors as

Example 1: −36 + 𝑟 2 𝑚 2 Example 2: −125 + 𝑛 4 𝑥 2