1. Factoring Polynomials We will learn how to factor cubic functions using factoring patterns. The factoring patterns we will use are the difference of two cubes and the sum of two cubes

2. 13=113=1 23=823=8 3 3 =27 4 3 =64 5 3 =125 6 3 =216 7 3 =343 8 3 =512 9 3 =729 10 3 =1000 Review: Perfect Cubes

3. Review: Other Cubes x3x3 =2 3 x 3 = 8x 3 8x38x3 (2x) 3 x6x6 =(x 2 ) 3 x6x6 We want to write cubes as a power with a base and an exponent (a) n

4. Are these cubes? 27x 3 9x39x3 x9x9 =3 3 x 3 =(3x) 3  9 is not a cube number =(x 3 ) 3

5. It is important to be able to recognize symbols, signs and patterns

6. Recognizing a Difference or Sum of Two Cubes Difference of Two Cubes Sum of Two Cubes a 3 – b 3 a 3 + b 3 The polynomial is a binomial a 3 – b 3 a 3 + b 3 Two terms Both terms are cubes a 3 – b 3 a 3 + b 3 The binomial is either a difference or a sum a3 – b3a3 – b3 a3 + b3a3 + b3

7. These are a difference and sum of cubes 64x 3 – 125y 3 x 3 + y 3 =4 3 x 3 – 5 3 y 3 =(4x) 3 – (5y) 3 =(x) 3 + (y) 3

8. These are not a difference or sum of cubes 72x 3 – y 3 x 3 + x 2 72 is not a cubex 2 is not a cube

9. Classify the Following Polynomials Difference of CubesSum of CubesNeither 3x 3 – 5x 2 +2 27y 3 +x 3 8x 2 – 4y 2 2x 3 – 4y 3 a 3 – 1 125 + y 3 27y 3 +x 3 3x 3 – 5x 2 +2 8x 2 – 4y 2 2x 3 – 4y 3 a 3 – 1 125 + y 3

10. a 3 – b 3 = (a – b)(a 2 + ab + b 2 ) a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) Use SOFAS to remember the factoring pattern. Difference of Two Cubes Sum of Two Cubes Square the first term Opposite sign First term times second term Always positive Square the second term

11. a 3 – b 3 = (a – b)(a 2 + ab + b 2 )a 3 + b 3 = (a + b)(a 2 – ab + b 2 ) Difference of Two CubesSum of Two Cubes 3 – 3 = ( – )( 2 + + 2 ) Recognize as difference or sum of cubes Identify a and b Substitute values into the factoring pattern Simplify How do I use the factoring patterns?

12. Examples = ( – )( 2 + + 2 ) 27x 3 – 8 Recognize as difference or sum of cubes Identify a and b =(3x) 3 – (2) 3 3x 222 Substitute values into the factoring pattern Simplify = (3x – 2)(9x 2 + 6x + 4)

13. Examples = ( + )( 2 – + 2 ) 8x 3 + 1 Recognize as difference or sum of cubes Identify a and b =(2x) 3 + (1) 3 2x 111 Substitute values into the factoring pattern Simplify = (2x + 1)(4x 2 – 2x + 1)

14. Examples = ( + )( 2 – + 2 ) 125x 3 + 216y 3 Recognize as difference or sum of cubes Identify a and b =(5x) 3 + (6y) 3 5x 6y Substitute values into the factoring pattern Simplify = (5x + 6y)(25x 2 – 30xy + 36y 2 )

15. Use Your Whiteboards 64x 3 + 125 x 3 – 27 343y 3 + x 3

16. Independent Practice