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# Factoring Trinomials

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1. Chapters 9.3 and 9.4 Factoring Trinomials

2. Factoring Trinomials • Lesson Objective: • Students will know how to use the box method to factor a trinomial

3. Factoring Trinomials Review • Example: Solve (x + 3)(x + 2) • Remember we use the box method to solve this problem

4. Factoring Trinomials Review • Solve: (x + 3)(x + 2) x +3 x x x2 * x x 3x * 3 6 2x * x 2 * 3 +2 2

5. Factoring Trinomials X + 3 +2 x x2 3x 2x 6 x2 + 5x + 6

6. Factoring Trinomials • Today we’re going to learn how to do this in reverse

7. Factoring Trinomials • Example 1: Factor x2 + 7x + 12 • We’re going to use the box method to factor this problem

8. Factoring Trinomials • Factor x2 + 7x + 12 • Usually we put the problem on the outside, but we were given the answer instead! • So we need to find the numbers on the outside

9. Factoring Trinomials • Factor x2 + 7x + 12 • In order to find our answer we had to take the numbers from inside the square + 7x + 12 X2

10. Factoring Trinomials • Factor x2 + 7x + 12 • Let’s put everything back into the box X2 12 + 7x + 12 X2

11. Factoring Trinomials • Factor x2 + 7x + 12 • As you can see, we have one number and 2 spots for it • We have to split the 7x into 2 numbers X2 12 + 7x + 12 X2

12. Factoring Trinomials • Factor x2 + 7x + 12 • Start by multiplying the 12 and x2 • = 12x2 X2 12 + 7x + 12 X2

13. Factoring Trinomials • We’re going to have to set up 2 tables

14. Factoring Trinomials • In the first table we put products that multiply to 12x2 1x 12x * 6x 2x * 3x 4x *

15. Factoring Trinomials • In the second table we add instead of multiply to get the number in the middle 1x 12x 1x + 12x * + 6x 2x 6x 2x * + 3x 4x 3x 4x *

16. Factoring Trinomials • Notice the 3x and 4x work for both tables 1x 12x 1x + 12x * + 6x 2x 6x 2x * + 3x 4x 3x 4x *

17. Factoring Trinomials • Therefore, these are the two numbers that fill in the box 1x 12x 1x + 12x * + 6x 2x 6x 2x * + 3x 4x 3x 4x *

18. Factoring Trinomials • It doesn’t matter where each one goes, so put them both in the box X2 3x 4x 12 + 7x + 12 X2

19. Factoring Trinomials x 3 • We can use the Greatest Common Factor to get the numbers on the outside • The GCF of x2 and 4x is x • The GCF of 3x and 12 is 3 x X2 3x 4 4x 12 + 7x + 12 X2

20. Factoring Trinomials x 3 • We can then put the numbers on top together for one parenthesis • The side is the other parenthesis x X2 3x 4 4x 12 (x + 4) (x + 3) + 7x + 12 = X2

21. Factoring Trinomials Let’s try that again! • Factor: x2 + 3x – 4 • Start with the box

22. Factoring Trinomials • Factor x2 + 3x – 4 • Let’s put everything back into the box X2 – 4 + 3x – 4 X2

23. Factoring Trinomials • Factor x2 + 3x + 12 • Start by multiplying the -4 and x2 • = -4x2 X2 – 4 + 3x – 4 X2

24. Factoring Trinomials • Set up your two tables 1x -4x 1x + -4x * + 4x -1x 4x -1x * + 2x -2x 2x -2x *

25. Factoring Trinomials • We see that -1x and 4x works for both tables so those are our numbers 1x -4x 1x + -4x * + 4x -1x 4x -1x * + 2x -2x 2x -2x *

26. Factoring Trinomials • It doesn’t matter where each one goes, so put them both in the box X2 -1x 4x -4 + 3x – 4 X2

27. Factoring Trinomials x -1 • We can use the Greatest Common Factor to get the numbers on the outside • The GCF of x2 and 4x is x • The GCF of -1x and -4 is -1 x X2 -1x 4 4x -4 + 3x – 4 X2

28. Factoring Trinomials x -1 • Always take the sign closest to the number on the outside! x X2 -1x 4 4x -4 + 3x – 4 X2

29. Factoring Trinomials x -1 • We can then put the numbers on top together for one parenthesis • The side is the other parenthesis x X2 -1x 4 4x -4 (x + 4) (x – 1) + 3x – 4 = X2

30. Factoring Trinomials Practice • Factor the following: 1. x2 + 8x + 12 2. x2 + 18x + 32 3. x2 – 4x + 4 4. x2 – 7x + 6 5. x2 + 10x + 25

31. Factoring Trinomials Practice • Factor the following: 1. x2 + 8x + 12 2. x2 + 18x + 32 (x + 6)(x + 2) (x + 16)(x + 2) 3. x2 – 4x + 4 4. x2 – 7x + 6 (x – 6)(x – 1) (x – 2)(x – 2) 5. x2 + 10x + 25 (x + 5)(x + 5)

32. Factoring Trinomials x2 + 6x = 7 • Solve: • If you see an x2 and an equals sign, you have to get everything on one side of the equation • Now we need to factor the left side -7-7 x2 + 6x – 7 = 0

33. Factoring Trinomials • Let’s put everything back into the box • Multiply -7 and x2 • = -7x2 X2 – 7 x2 + 6x – 7 = 0

34. Factoring Trinomials • Set up your two tables -1x 7x -1x + 7x *

35. Factoring Trinomials • We see that -1x and 7x works for both tables so those are our numbers -1x 7x -1x + 7x *

36. Factoring Trinomials • Plug in the two numbers X2 -1x – 7 7x x2 + 6x – 7 = 0

37. Factoring Trinomials x -1 • Find the GCF to put on the outside of the box x X2 -1x 7 – 7 7x x2 + 6x – 7 = 0

38. Factoring Trinomials x -1 • Replace the equation with your answer x X2 -1x 7 – 7 7x (x + 7) (x – 1) x2 + 6x – 7 = 0

39. Factoring Trinomials (x + 7) (x – 1) = 0 • Just a reminder: x*y = 0 means that either x or y has to be zero! • We must set both parenthesis equal to zero and solve x + 7 = 0 x – 1 = 0 -7-7 +1+1 x = -7 x = 1

40. Factoring Trinomials Practice • Factor the following: 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0 5. x2 + 10x – 24 = 0

41. Factoring Trinomials Practice • Factor the following: 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16 x = -3 and -4 x = -8 or -2 3. x2 + 6 = 5x 4. x2 – 5x – 6 = 0 x = 6 or -1 x = 2 or 3 5. x2 + 10x – 24 = 0 x = -12 or 2

42. Factoring Trinomials 2x2 + 15x + 18 • Example 4: Factor • We’re going to work this like the other problems

43. Factoring Trinomials • Start with the box! • Multiply 18 and 2x2 • = 36x2 2x2 18 2x2 + 15x + 18

44. Factoring Trinomials • Set up your two tables 1x 36x 1x 36x + * 2x 18x 2x 18x + * 3x 12x 3x 12x + *

45. Factoring Trinomials • 3x and 4x works for both, so those are our numbers 1x 36x 1x + 36x * 2x 18x 2x 18x + * 3x 12x 3x 12x + *

46. Factoring Trinomials • Plug in the two numbers 2x2 12x 18 3x 2x2 + 15x + 18

47. Factoring Trinomials x 6 • Find the GCF to put on the outside of the box 2x 2x2 12x 3 18 3x 2x2 + 15x + 18

48. Factoring Trinomials x 6 • We can then put the numbers on top together for one parenthesis • The side is the other parenthesis 2x 2x2 12x 3 18 3x 2x2 + 15x + 18 (2x + 3) (x + 6)

49. Factoring Trinomials • Example 5: Factor: • Plug them into the box • Multiply -6 and 2x2 • = -12x2 2x2 + 3x – 6 2x2 -6 2x2 + 3x – 6

50. Factoring Trinomials • Set up your two tables -1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + *