1
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2
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We're asked to factor 35k squared plus 100k, minus 15.

3
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And because we have a non-1 coefficient out here, the best

4
00:00:10,929 --> 00:00:14,599
thing to do is probably to factor this by grouping.

5
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But before we even do that, let's see if there's a common

6
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factor across all of these terms, and maybe we can get a

7
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1 coefficient, out there.

8
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If we can't get a 1 coefficient, we'll at least

9
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have a lower coefficient here.

10
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And if we look at all of these numbers, they all look

11
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divisible by 5.

12
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In fact their greatest common factor is 5.

13
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So let's at least factor out a 5.

14
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So this is equal to 5 times-- 35k squared

15
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divided by 5 is 7k squared.

16
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100k divided by 5 is 20k.

17
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And then negative 15 divided by 5 is negative 3.

18
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So we were able to factor out a 5, but we still don't have a

19
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1 coefficient here, so we're still going to have to factor

20
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by grouping.

21
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But at least the numbers here are smaller so it'll be easier

22
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to think about it in terms of finding numbers whose product

23
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is equal to 7 times negative 3, and whose

24
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sum is equal to 20.

25
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So let's think about that.

26
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Let's figure out two numbers that if I were to add them, or

27
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even better if I were to take their product, I get 7 times

28
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negative 3, which is equal to negative 21.

29
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And if I were to take their sum, if I add those two

30
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numbers, it needs to be equal to 20.

31
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Now, once again, because their product is a negative number,

32
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that means they have to be of different signs, so when you

33
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add numbers of different signs, you could view it as

34
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you're taking the difference of the positive versions.

35
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So the difference between the positive versions of the

36
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number has to be 20.

37
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So the number that immediately jumps out is we're probably

38
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going to be dealing with 20 and 21, and 1 will be the

39
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negative, because we want to get to a positive 20.

40
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So let's think about it.

41
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So if we think of 20 and negative 1, their product is

42
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negative 21.

43
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Sorry.

44
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If we take 21 and negative 1, their product is negative 21.

45
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21 times negative 1 is negative 21.

46
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and if you take their sum, 21 plus negative 1,

47
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that is equal to 20.

48
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So these two numbers right there fit the bill.

49
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Now, let's break up this 20k right here into a 21k and a

50
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negative 1k.

51
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So let's do that.

52
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So let's rewrite the whole thing.

53
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We have 5 times 7k squared, and I'm going to break this

54
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20k into a-- let me do it in this color right here-- I'm

55
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going to break that 20k into a plus 21k, minus k.

56
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Or you could say minus 1k if you want.

57
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I'm using those two factors to break it up.

58
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And then we finally have the minus 3 right there.

59
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Now, the whole point of doing that is so that we can now

60
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factor each of the two groups.

61
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This could be our first group right here.

62
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And so what can we factor out of that group right there?

63
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Well, both of these are divisible by 7k, so we can

64
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write this as 7k times-- 7k squared divided by 7k, you're

65
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just going to have a k left over.

66
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And then plus 21k divided by 7k is just going to be a 3.

67
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So that factors into that.

68
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And then we can look at this group right here.

69
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They have a common factor.

70
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Well, we can factor out a negative 1 if we like, so this

71
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is equal to negative 1 times-- k divided by negative 1 is k.

72
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Negative 3 divided by negative 1 is positive 3.

73
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And, of course, we have this 5 sitting out there.

74
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75
00:03:51,849 --> 00:03:55,079
Now, ignoring that 5 for a second, you see that both of

76
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these inside terms have k plus 3 as a factor.

77
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So we can factor that out.

78
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So let's ignore this 5 for a second.

79
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This inside part right here, the stuff that's inside the

80
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parentheses, we can factor k plus 3 out, and it becomes k

81
00:04:13,259 --> 00:04:24,980
plus 3, times k plus 3, times 7k minus 1.

82
00:04:24,980 --> 00:04:28,110
And if this seems a little bizarre to you, distribute the

83
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k plus 3 on to this.

84
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K plus 3 times 7k is that term, k plus 3 times negative

85
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1 is that term.

86
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And, of course, the whole time you have

87
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that 5 sitting outside.

88
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You have that 5.

89
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We don't even have to put parentheses there.

90
00:04:38,649 --> 00:04:42,769
5 times k plus 3, times 7k minus 1.

91
00:04:42,769 --> 00:04:45,049
And we factored it, we're done.

92
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