1
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2
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We need to factor negative 12f squared minus 38f, plus 22.

3
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So a good place to start is just to see if, is there any

4
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common factor for all three of these terms?

5
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When we look at them, they're all even.

6
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And we don't like a negative number out here.

7
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So let's divide everything, or let's factor out a negative 2.

8
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So this expression right here is the same thing as negative

9
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2 times-- what's negative 12f squared divided by negative 2?

10
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It's positive 6f squared.

11
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Negative 38 divided by negative 2 is positive 19, so

12
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it'll be positive 19f.

13
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And then 22 divided by negative 22-- oh, sorry, 22

14
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divided by negative 2 is negative 11.

15
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So we've simplified it a bit.

16
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We have the 6f squared plus 19f, minus 11.

17
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We'll just focus on that part right now.

18
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And the best way to factor this thing, since we don't

19
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have a 1 here as the coefficient on the f squared,

20
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is to factor it by grouping.

21
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So we need to look for two numbers whose product is 6

22
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times negative 11.

23
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So two numbers, so a times b, needs to be equal to 6 times

24
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negative 11, or negative 66.

25
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And a plus b needs to be equal to 19.

26
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So let's try a few numbers here.

27
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So let's see, 22, I'm just thinking of numbers that are

28
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roughly 19 apart, because they're going to be of

29
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different signs.

30
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So 22 and 3, I think will work.

31
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Right.

32
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If we take 22 times negative 3, that is negative 66, and 22

33
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plus negative 3 is equal to 19.

34
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And the way I kind of got pretty close to this number

35
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is, well, you know, they're going to be of different

36
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signs, so the positive versions of them have to be

37
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about 19 apart, and that worked out.

38
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22 and negative 3.

39
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So now we can rewrite this 19f right here as the sum of

40
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negative 3f and 22f.

41
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That's the same thing as 19f.

42
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I just kind of broke it apart.

43
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And, of course, we have the 6f squared and we have

44
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the minus 11 here.

45
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Now, you're probably saying, hey Sal, why did you put the

46
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22 here and the negative 3 there?

47
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Why didn't you do it the other way around?

48
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Why didn't you put the 22 and then the negative 3 there?

49
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And my main motivation for doing it, I like to put the

50
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negative 3 on the same side with the 6 because they have

51
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the common factor of the 3.

52
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I like to put the 22 with the negative 11, they have the

53
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same common factor of 11.

54
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So that's why I decided to do it that way.

55
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So now let's do the grouping.

56
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And, of course, you can't forget this negative 2 that we

57
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have sitting out here the whole time.

58
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So let me put that negative 2 out there, but that'll just

59
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kind of hang out for awhile.

60
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But let's do some grouping.

61
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So let's group these first two.

62
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And then we're going to group this-- let me get a nice color

63
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here-- and then we're going to group this second two.

64
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Well, that's almost an identical color.

65
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Let me do it in this purple color.

66
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And then we can group that second two right there.

67
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So these first two, we could factor out a negative 3f, so

68
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it's negative 3f times-- 6f squared divided by negative 3f

69
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is negative 2f.

70
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71
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And then negative 3f divided by negative 3f is just

72
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positive f.

73
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Actually, a better way to start, instead of factoring

74
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out a negative 3f, let's just factor out 3f, so we don't

75
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have a negative out here.

76
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We could do it either way.

77
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But if we just factor out a 3f, 6f squared

78
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divided by 3f is 2f.

79
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And then negative 3f divided by 3f is negative 1.

80
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So that's what that factors into.

81
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And then that second part, in that dark purple color, can

82
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factor out an 11.

83
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And if we factor that out, 22f divided by 11 is 2f, and

84
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negative 11 divided by 11 is negative 1.

85
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And, of course, once again, you have that negative 2

86
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hanging out there.

87
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88
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Now, inside the parentheses, we have two terms, both of

89
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which have 2f minus 1 as a factor.

90
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So we can factor that out.

91
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This whole thing is just an exercise in doing the reverse

92
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distributive property, if you will.

93
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So let's factor that out, so you have 2f minus 1, times

94
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this 3f, and then times that plus 11.

95
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Let me do that in the same shade of

96
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purple right over there.

97
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And you know, you can distribute it if you like.

98
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2f minus 1 times 3f will give you this term, 2f f minus 1

99
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times 11 will give you that term.

100
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And we can't forget that we still have that negative 2

101
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hanging out outside.

102
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I want to change the colors on it.

103
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104
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And we're done factoring it.

105
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Negative 12f squared minus 38f, plus 22 is negative 2

106
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times 2f minus 1, times 3f plus 11.

107
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