1. 1. The height of an object launched t seconds is modeled by h(t) = -16t 2 + 32t + 25. Find the vertex and interpret what it means. What is the height of the object after 1.5 seconds? 2. The table below shows the average sale price p of a house in Suffolk County, Massachusetts for various years since 1988. Use your graphing calculator to find a quadratic model for this data. If this trend continues, what would the cost of a house be in 2010? Algebra II 1

2. Factoring Quadratics Algebra II

3.  Greatest Common Factor  Trinomials with leading coefficient of 1  x 2 + bx + c  Trinomials with leading coefficient other than 1  ax 2 + bx + c  Difference of Two Squares  Four term polynomial – factor by grouping Algebra II 3

4. Take out the greatest common factor of a trinomial by dividing each term by the GCF (greatest common factor) Examples: 1. 16x 3 – 12x 2 + 4x 2. 15xy 2 – 25x 2 y GCF: 4x 4x(4x 2 – 3x + 1) GCF: 5xy 5xy(3y – 5x) Algebra II 4

5. 3. 27m 3 p 2 + 9mp - 54p 2 4. 10x – 40y GCF: 9p 9p(3m 3 p + m – 6p) GCF: 10 10(x – 4y) Algebra II 5

6. x 2 + bx + c = (x + )(x + ) The product of these numbers is c. The sum of these numbers is b. Algebra II 6

7. 1. x 2 – 12x – 28 2. x 2 + 3x – 10 3. x 2 + 12x + 35 4. y 2 – 10y – 24 5. x 2 – 6x + 10 6. p 2 + 3p – 40 Algebra II 7 (x – 14)(x + 2) (x + 5)(x – 2) (x + 7)(x + 5) (y – 12)(x + 2) prime (p + 8)(p – 5)

8.  You should always check your factoring results by multiplying the factored polynomial to verify that it is equal to the original polynomial.  You can detect computational errors or errors in the signs of your numbers by checking your results. 8 Algebra II

9. How are we going to factor if the leading coefficient is not 1?  The “X” Method ax 2 + bx + c 9 Algebra II a  c b the “#s” are factors of a  c that add up to b #1#1 #2#2

10.  It is actually a graphic organization of “guess & check”  The “#s” are not what go in the binomials  Completely unnecessary if the leading coefficient is 1 10 Algebra II

11. ( ) Factor 8x 2 – 14x + 5 Algebra II 11 4x – 52x – 1 40 -14 -4-10 1818 4242 1515

12. ( ) Factor 6x 2 – 11x – 10 Algebra II 12 3x + 22x – 5 -60 -11 -154 1616 3232 1  10 2525

13. Factor 6x 2 – 2x – 20 2(3x 2 – x – 10) 2 ( )( ) Algebra II 13 3x + 5 x – 2 -30 -65 3131 1  10 2525

14. ( ) Factor 21x 2 – 13x + 2 Algebra II 14 3x – 17x – 2 42 -13 -6-7 1  21 3737 1212

15. Factor 10a 3 + 17a 2 +3a a(10a 2 + 17a + 3) a ( )( ) Algebra II 15 2a + 35a + 1 30 17 215 1  10 1313 2525

16. ( ) Factor 8x 2 – x – 9 Algebra II 16 8x – 9 x + 1 -72 8-9 1818 2424 1919 3333

17. Factor 4y 2 – 2y – 12 2(2y 2 – y – 6) 2 ( )( ) Algebra II 17 2y + 3 y – 2 -12 -43 2121 1616 2323

18. Factor 45a 2 + 57a – 30 3(15a 2 +19a – 10) 3 ( )( ) Algebra II 18 3a + 55a – 2 -150 19 -625 1  15 1  10 2525 3535

19. ( ) Factor 15x 2 + 11x + 2 Algebra II 19 3x + 15x + 2 30 11 65 1  15 3535 1212

20. ( ) Factor 15x 2 – 29x – 2 Algebra II 20 15x + 1 x – 2 -30 -29 -301 1  15 3535 1212

21. 11. 3x 2 – 17x + 10 12. 4x 2 – 4x – 3 13. 49x 2 – 14x + 1 14. 16y 2 + 4y + 1 15. 5x 2 + 17x + 14 16. 3p 2 + p - 10 (3x – 2)(x – 5) (2x – 3)(2x + 1) (7x – 1)(7x – 1) prime (5x + 7)(x + 2) (3p – 5)(p + 2) Algebra II 21

22. 17. 8x 2 – 29x – 12 18. 12x 2 + 19x + 5 19. 4x 2 – 10x + 3 20. 16y 2 + 2y – 3 21. 9x 2 + 12x + 4 22. 6p 2 – 13p + 5 (x – 4)(8x + 3) (3x + 1)(4x + 5) prime (2y + 1)(8y – 3) (3x + 2)(3x + 2) (2p – 1)(3p – 5) Algebra II 22

23. Is it the difference of two Squares? Factoring the Difference of Two Squares a 2 – b 2 = (a + b)(a – b) Algebra II 23

24. Factor: x 2 – 9 (x – 3)(x + 3) Algebra II 24

25. 1. 4x 2 – 9 2. 9x 2 – 1 3. 16x 2 + 25 4. 1 – 25y 2 5. 49y 4 – 9z 2 6. 81p 2 – 25 (2x – 3)(2x + 3) (3x – 1)(3x + 1) NOT A DIFF. (1 – 5y)(1 + 5y) (7y 2 – 3z)(7y 2 + 3z) (9p – 5)(9p + 5) Algebra II 25 prime

26. 1. 3x 2 – 27 2. 4x 2 + 4x – 8 3. 5x 2 – 20 4. 14x 2 + 2x - 12 3(x 2 – 9) 3(x – 3)(x + 3) 4(x 2 + x – 2) 4(x – 1)(x + 2) 5(x 2 – 4) 5(x – 2)(x + 2) 2(7x 2 + x – 6) 2(7x – 6)(x + 1) Algebra II 26

27. 5. 2u 2 + 8u 6. 10x 2 + 34x + 28 7. 4x 4 – 64x 2 8. 30x 2 – 57x + 21 2u(u + 4) 2(5x 2 + 17x + 14) 2(5x + 7)(x + 2) 4x 2 (x 2 – 16) 4x 2 (x – 4)(x + 4) 3(10x 2 – 19x + 7) 3(2x – 1)(5x – 7) Algebra II 27

28.  If a polynomial has for terms:  Factor the GCF out of the first two terms  Factor the GCF out of the second two terms  Factor out the common binomial and write as two binomials 28 Algebra II

29. 1. 29 Algebra II