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

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

5. Factoring Trinomials X 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 X2
+ 7x
+ 12

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

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 X2
+ 7x
+ 12

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

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

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

15. 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials
In the second table we add instead of multiply to get the number in the middle
Multiplies To
12x2
Adds To 7x 1x
12x 1x +
12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x *

16. 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials
Notice the 3x and 4x work for both tables
Multiplies To
12x2
Adds To 7x 1x
12x 1x +
12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x *

17. 1x 12x 1x + 12x * 2x + 6x 2x 6x * 3x + 4x 3x 4x * Factoring Trinomials
Therefore, these are the two numbers that fill in the box
Multiplies To
12x2
Adds To 7x 1x
12x 1x +
12x * 2x + 6x 2x 6x * 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 X2
+ 7x
+ 12

19. x 3 x 4 Factoring Trinomials
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 X2
+ 7x
+ 12

20. x 3 x 4 (x + 3) (x + 4) Factoring Trinomials
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 + 3)
(x + 4) X2
+ 7x
+ 12 =

21. Let’s try that again! Factoring Trinomials 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 X2
+ 3x
– 4

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

24. 1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x *
Factoring Trinomials
Set up your two tables
Multiplies To
-4x2
Adds To 3x 1x
-4x 1x +
-4x *
-1x + 4x
-1x 4x * 2x +
-2x 2x
-2x *

25. 1x -4x 1x + -4x * -1x + 4x -1x 4x * 2x + -2x 2x -2x *
Factoring Trinomials
We see that -1x and 4x works for both tables so those are our numbers
Multiplies To
-4x2
Adds To 3x 1x
-4x 1x +
-4x *
-1x + 4x
-1x 4x * 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 X2
+ 3x
– 4

27. x -1 x 4 Factoring Trinomials
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 X2
+ 3x
– 4

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

29. x -1 x 4 (x – 1) (x + 4) Factoring Trinomials
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 – 1)
(x + 4) X2
+ 3x
– 4 =

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

31. Practice Factoring Trinomials 1. x2 + 8x + 12 2. x2 + 18x + 32
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 – 2)(x – 2)
(x – 6)(x – 1)
5. x2 + 10x + 25
(x + 5)(x + 5)

32. x2 + 6x = 7 -7 -7 x2 + 6x – 7 = 0 Factoring Trinomials 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
x2 + 6x – 7 = 0

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

34. -1x 7x -1x + 7x * Factoring Trinomials Set up your two tables
Multiplies To
-7x2
Adds To 6x
-1x 7x
-1x + 7x *

35. -1x 7x -1x + 7x * Factoring Trinomials
We see that -1x and 7x works for both tables so those are our numbers
Multiplies To
-7x2
Adds To 6x
-1x 7x
-1x + 7x *

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

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

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

39. (x – 1) (x + 7) = 0 x + 7 = 0 x – 1 = 0 -7 -7 +1 +1 x = -7 x = 1
Factoring Trinomials
(x – 1)
(x + 7)
= 0
x + 7
= 0
x – 1
= 0
+1 +1
x = -7
x = 1
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

40. Practice Factoring Trinomials 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16
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. Practice Factoring Trinomials 1. x2 + 7x + 12 = 0 2. x2 + 10x = -16
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 = 2 or 3
x = 6 or -1
5. x2 + 10x – 24 = 0
x = -12 or 2

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

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

44. Factoring Trinomials
Set up your two tables
Multiplies To
36x2
Adds To
15x 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
Multiplies To
36x2
Adds To
15x 1x
36x 1x +
36x * 2x
18x 2x
18x + * 3x
12x 3x
12x + *

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

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

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

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

50. -1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + *
Factoring Trinomials
Set up your two tables
Multiplies To
-12x2
Adds To 3x
-1x
12x
-1x +
12x *
-2x 6x
-2x 6x + *
-3x 4x
-3x 4x + *

51. -1x 12x -1x + 12x * -2x 6x -2x 6x + * -3x 4x -3x 4x + *
Factoring Trinomials
No factors work, so we can’t factor this equation
Multiplies To
-12x2
Adds To 3x
-1x
12x
-1x +
12x *
-2x 6x
-2x 6x + *
-3x 4x
-3x 4x + *

52. 2x2 + 3x – 6 Prime Factoring Trinomials
Since we can’t factor this problem we call it
Prime

53. Practice Factoring Trinomials 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2
Factor the following:
1. 2x2 + 5x + 2
2. 3x2 – 7x + 2
3. 4x2 + 8x – 5
4. 4x2 – 3x – 3
5. 6x2 – 13x + 6

54. Practice Factoring Trinomials 1. 2x2 + 5x + 2 2. 3x2 – 7x + 2
Factor the following:
1. 2x2 + 5x + 2
2. 3x2 – 7x + 2
(x + 2)(2x + 1)
(3x – 1)(x – 2)
3. 4x2 + 8x – 5
4. 4x2 – 3x – 3
(2x + 5)(2x – 1)
Prime
5. 6x2 – 13x + 6
(3x – 2)(2x – 3)

55. 12x2 – 32x – 12 Factoring Trinomials Example 6: Factor
The first thing we should do is look for a common factor
This equation has a common factor
The GCF is 4
12x2 – 32x – 12

56. 12x2 – 32x – 12 ___ ___ __ 4 4 4 (3x2 – 8x – 3) 4 Factoring Trinomials
Example 5: Factor
Factor out the 4
12x2 – 32x – 12
___ ___ __
(3x2
– 8x
– 3) 4

57. 4(3x2 – 8x – 3) Factoring Trinomials
Factor what’s inside the parenthesis, ignore the 4
Plug into the box
Multiply -3 and 3x2
= -9x2
3x2 -3
4(3x2 – 8x – 3)

58. 1x -9x 1x + -9x * Factoring Trinomials Set up your two tables
Multiplies To
-9x2
Adds To
-8x 1x
-9x 1x +
-9x *

59. 1x -9x 1x + -9x * Factoring Trinomials
1x and -9x works for both, so those are our numbers
Multiplies To
-9x2
Adds To
-8x 1x
-9x 1x +
-9x *

60. 4(3x2 – 8x – 3) Factoring Trinomials Plug in the two numbers 3x2 1x-3
4(3x2 – 8x – 3)

61. 3x 1 x -3 4(3x2 – 8x – 3) Factoring Trinomials
Find the GCF to put on the outside of the box x
3x2 1x -3
-9x -3
4(3x2 – 8x – 3)

62. 3x 1 x -3 4(3x2 – 8x – 3) (3x + 1) (x – 3) 4 Factoring Trinomials
Find the GCF to put on the outside of the box
Put the 4 back in front x
3x2 1x -3
-9x -3
4(3x2 – 8x – 3)
(3x + 1)
(x – 3) 4

63. Practice Factoring Trinomials 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6
Factor the following:
1. 4x2 + 10x + 4
2. 9x2 – 21x + 6
3. 20x2 + 40x – 25
4. 18x3 + 15x2 – 18x
5. 36x3 – 78x2 + 36x

64. Practice Factoring Trinomials 1. 4x2 + 10x + 4 2. 9x2 – 21x + 6
Factor the following:
1. 4x2 + 10x + 4
2. 9x2 – 21x + 6
2(x + 2)(2x + 1)
3(3x – 1)(x – 2)
3. 20x2 + 40x – 25
4. 18x3 + 15x2 – 18x
5(2x + 5)(2x – 1)
3x(2x+3)(3x-2)
5. 36x3 – 78x2 + 36x
6x(3x – 2)(2x – 3)