1. For Educational Use Only © 2010 10.8 Factoring Using the Distributive Property Brian Preston Algebra 1 2009-2010

2. For Educational Use Only © 2010 How fast do you need to run so this does not happen? Real World Application

3. For Educational Use Only © 2010 1) Use the distributive property to factor a polynomial. 2) Solve polynomial equations by factoring. Lesson Objectives

4. For Educational Use Only © 2010 Review 1) Factor the GCF out of 14x 4 – 21x 2. Taking the GCF out is always out first step. Sometimes there will be no GCF.

5. For Educational Use Only © 2010 Review 1) What is the GCF of 14x 4 – 21x 2. 14x 4 21x 2 GCF = = = 2  7 3  7 7  x  x  x = 7x 2  x

6. For Educational Use Only © 2010 Review 7x 2 ( 2 x2x2 – 3 ) 1) Factor the GCF out of 14x 4 – 21x 2.

7. For Educational Use Only © 2010 (x) 2 2 2 2x Rule If you take out a GCF, always look to factor again. ( x x 2) 3x 2 – 12 2 )( 2 ) = 4 3 ( + – = – x2x2 – ) 3 [ ] 3 x

8. For Educational Use Only © 2010 x2x2 3) 4x 3 + 20x 2 + 24x Factor the expression completely. 4 ( ) + 5 + 6 Example First, factor the GCF out. Then, factor x 2 + 5x + 6 normally. Last, make sure the GCF is a part of the answer. x x

9. For Educational Use Only © 2010 x 2 + 5x + 6 – 1 + 6 Example 3) Factors of 6 1  6 + 5 = 2  3 Factor the trinomial. 4x ( ) 1 + 6 7 = 6 – 1 5 =

10. For Educational Use Only © 2010 Rule x 2 – 2x – 8 One (-) & One (+) Patterns for factoring trinomials. (x + 2) (x - 4) x 2 + 6x + 8 (x + 2) (x + 4) x 2 – 6x + 8 (x – 2) (x – 4) Two (+) Two (-) x 2 + 2x – 8 (x + 4) (x - 2)

11. For Educational Use Only © 2010 4x ( x 2 + 5x + 6 3) ) 2 + 3 Example +5 = Factor the trinomial. Factors of 6 1  6 2  3 2 + 3 5 = 3 – 2 1 =

12. For Educational Use Only © 2010 1x +2 +3 +2 +3 4x ( x 2 + 5x + 6 3) ) 2 + 3 3 2 ( + 3 + 2 Example +5 = Factor the trinomial. Factors of 6 )( ) x2x2 1  6 2  3 4x

13. For Educational Use Only © 2010 4) In the sport of pole-vaulting, the height h (in feet) reached by a pole-vaulter is a function of v, the velocity of the pole-vaulter, as shown in the model below. The constant g is approximately 32 feet per second per second. To reach a height of 9 feet, what is the pole- vaulter’s velocity? velocity 99 32 (32) Real World Application 9 v h = 2 2 g

14. For Educational Use Only © 2010 4) 9 = What is the pole-vaulter’s velocity needed to reach 9 feet high? = 9 Real World Application v2v2 2(32) v2v2 64 1 

15. For Educational Use Only © 2010 64 1 4) 9 = What is the pole-vaulter’s velocity needed to reach 9 feet high? = 9 = Real World Application v2v2 2(32) v2v2 64  1  276v2v2 = 24 ft/secv + - 64 1

16. For Educational Use Only © 2010 What is the pole-vaulter’s velocity needed to reach 9 feet high? 24 ft/sec Real World Application

17. For Educational Use Only © 2010 m2m2 5) – 7m 2 + 28m – 21 Factor the expression completely. – 7 ( ) – 4 + 3 Example First, factor the GCF out. Then, factor x 2 + 5x + 6 normally. Last, make sure the GCF is a part of the answer. m

18. For Educational Use Only © 2010 m 2 – 4m + 3 – 1 + – 3 5) Factors of 3 1  3 – 4 -4 = Factor the expression completely. – 7 ( ) Example 1 + 3 4 = 3 – 1 2 =  (–1) to all

19. For Educational Use Only © 2010 1m -3 -3 m 2 – 4m + 3 5) – 7 ( ) – 1 + – 3 – 3 – 1 ( – 3 – 1 -4 = Factor the expression completely. )( ) m2m2 1  3 – 7 Factors of 3 Example

20. For Educational Use Only © 2010 (3x) 2 3x 2 2 2 Rule Factor the expression completely. ( 3x 6) 45x 4 – 20x 2 2 )( 2 ) = 4 5 ( + – = – 9 – ) 5x 2 [ ] x2x2 x2x2

21. For Educational Use Only © 2010 6262 6 6 (t) 2 t t Rule Factor the expression completely. ( t t 7) 4t 3 – 144t 6 )( 6 ) = 36 4 ( + – = – t2t2 – ) 4t [ ] t

22. For Educational Use Only © 2010 8) x 3 + 2x 2 + 3x + 6 Factor the expression by grouping. Example x x2x2 ( ) + 2 x 3 + 2x 2 + 3x + 6 ( )( ) x + 3 ( ) + 2 x ( ) x2x2 ( ) + 3 x ( ) + 2x ( ) Always try to factor again. Impossible

23. For Educational Use Only © 2010 – 9 x2x2 ) – 2 9) x 3 – 2x 2 – 9x + 18 Factor the expression by grouping. Example x x2x2 ( ) x 3 – 2x 2 – 9x + 18 ( )( ) x – 9 ( ) x ( ) – 2 ( x ( ) x ( ) Always try to factor again. ( x x 3 )( 3 ) + – ( x2 ) –

24. For Educational Use Only © 2010 The order to factoring. First, factor the GCF out. Then, identify which of the three factoring ways. (Two terms, three terms or four terms.) Rule Always try to factor again.

25. For Educational Use Only © 2010 3232 (2x) 2 3 2x3 Example Solve. ( 2x 10) 8x 3 – 18x = 0 3 )( 3 ) 9 2 ( + – – 4 – ) 2x [ ] x x2x2 = = = 0 0 0

26. For Educational Use Only © 2010 2 2 + 3 2 2 2 2 (2x – 3) (2x + 3) 2x (2x – 3) 2x (2x + 3) – 3 10) = 0 2x – 3 2x + 3 – 3 ( ) ( ) + 3 = 0 2x( ) = – 3 2x = 3 2 x= 2 x= 3 2 3 2 – Example Solve. 2 x= 0

27. For Educational Use Only © 2010 3 11) – 6x 2 + 34x + 56 = 0 Factor the expression completely. – 2 ( ) – 17 – 28 Example First, factor the GCF out. Then, factor the three- term way. x x2x2 = 0

28. For Educational Use Only © 2010 Example 11) – 2(3x 2 – 17x – 28) = 0 Factors of 84 1  84 – 17 -17 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 1 + 84 85 = 84 – 1 83 =

29. For Educational Use Only © 2010 11) – 2(3x 2 – 17x – 28) = 0 Example Factors of 84 1  84 -17 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 2 + 42 44 = 42 – 2 40 =

30. For Educational Use Only © 2010 11) – 2(3x 2 – 17x – 28) = 0 Example Factors of 84 1  84 -17 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 3 + 28 31 = 28 – 3 25 =

31. For Educational Use Only © 2010 11) – 2(3x 2 – 17x – 28) = 0 4 + – 21 Example -17 = Solve the equation by factoring. Factors of 84 1  84 2  42 3  28 4  21 6  14 7  12 4 + 21 25 = 21 – 4 17 =  (–1) to all

32. For Educational Use Only © 2010 1x 3x 1x3x +4 -21 ( 11) – 2(3x 2 – 17x – 28) = 0 4 + – 21 – 21 4 ( + 4 – 7 Example -17 = Solve the equation by factoring. Factors of 84 ) )=0 3x 2 1  84 2  42 3  28 4  21 6  14 7  12 – 2

33. For Educational Use Only © 2010 3 3 – 4 + 7 (x - 7) (3x + 4) (x - 7) (3x + 4) – 2 11) = 0 x – 7 3x + 4 – 4 ( ) ( ) + 7 = 0 –2( ) 3x = – 4 x = 7 3 x= 4 3 – Example Solve.

34. For Educational Use Only © 2010 The order to factoring. First, factor the GCF out. Then, identify which of the three factoring ways. (Two terms, three terms or four terms.) Rule Always try to factor again.

35. For Educational Use Only © 2010 Key Points & Don’t Forget 1) Step one: take the GCF out. 2) Don’t forget the negative signs. 3) Make sure the expressions or equations are in standard form before factoring. 4) Always try to factor again.

36. For Educational Use Only © 2010 pg. 462-463 #’s 10-37, 40-46 even The Assignment

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