1
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Factor 8k squared minus 24k minus 144.

2
00:00:07,707 --> 00:00:10,895
Now the first thing we can do here, just eyeballing each of these terms

3
00:00:10,895 --> 00:00:13,742
if we want to simplify it a good bit is

4
00:00:13,742 --> 00:00:16,528
all of these terms are divisible by 8.

5
00:00:16,528 --> 00:00:19,769
Clearly 8k squared is divisible by 8, 24 is divisible by 8,

6
00:00:19,769 --> 00:00:24,266
and 144, it might not be as obvious is divisble by 8,

7
00:00:24,266 --> 00:00:26,801
but it looks like it is. 8 goes into

8
00:00:26,801 --> 00:00:32,024
144- 8 goes into 14 one time, 1 times 8 is 8

9
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subtract, you get a 6 (14 minus 8 is 6),

10
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bring down the four- 8 goes into 64 8 times.

11
00:00:40,083 --> 00:00:42,846
So it goes into 144 18 times. So let's just

12
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factor out an 8 of this, and then that'll simplify

13
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our expression, it'll actually give us a leading one coefficient.

14
00:00:48,957 --> 00:00:54,102
So this will become 8 times k squared minus

15
00:00:54,102 --> 00:01:04,607
24 divided by 8 is three k minus 18.

16
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Now we have to factor this business in here,

17
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and remember if anything has the form

18
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x squared plus bx plus c, where you have a

19
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leading one coefficient (this is implicitly a

20
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one, we have that here in this expression in

21
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parentheses), then we literally just need a

22
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(and we can do this multiple ways) but we need

23
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to find two numbers whose sum is equal to the

24
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coefficient on x, so two numbers whose sum is equal

25
00:01:31,063 --> 00:01:35,642
to negative three, and whose product is equal

26
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to the constant term, and whose product is

27
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equal to negative 18. So let's just think

28
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about the factors of -18 here, let's see if we

29
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can do something interesting. So it could be

30
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one, and since it's negative one of the numbers

31
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has to be positive and one has to be negative.

32
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1 and 18 is if it was positive, and one of these

33
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could be positive and one could be negative,

34
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but no matter what, we're either going to have

35
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pos- we're either- if you, if this is negative

36
00:02:01,750 --> 00:02:05,365
and this is positive, they add up to 17; but

37
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if you switch them then they add up to negative 17,

38
00:02:07,248 --> 00:02:09,584
so those won't work. So either (you can write

39
00:02:09,584 --> 00:02:12,860
it this way) positive or negative 1 and then

40
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negative or positive 18 to show that they

41
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have to be different signs. So those don't work.

42
00:02:16,644 --> 00:02:20,049
Then you have positive or negative 3 and then

43
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negative or positive 6, just to know that they are

44
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different signs. So if you have positive 3 and

45
00:02:25,199 --> 00:02:28,116
negative 6, they add up to negative 3, which

46
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is what we need them to add up to. And clearly,

47
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they- positive 3 and negative 6 their product

48
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is negative 18. So it works! So we're going to go with

49
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positive 3 and negative 6 as our two numbers.

50
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Now, for this example, just for the sake of this

51
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example, we'll do this by grouping. So what we

52
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can do is we can separate this middle term right here

53
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as the sum of 3k and -6k, so I can write the

54
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negative 3k as plus 3k minus 6k (let me write the

55
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rest of it)- so we have k squared up here, plus 3k

56
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minus 6k, which is the same as this thing over here,

57
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and we have minus 18. And then all of that's being

58
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multiplied by 8. Now we're ready to group this thing-

59
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we can group these first two terms, they are both

60
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divisible by k, and then we can group these second

61
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two terms. So then we have 8 times (let me write

62
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brackets here, since we're using double parentheses

63
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- brackets are really just parentheses that look a

64
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little bit more serious)- OK, now let's factor

65
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out a k from this term right here (using a different

66
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color)- let's factor out the k here. So this is k

67
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times k plus 3, and then we have a plus, and over

68
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here it looks like we could factor out a negative 6.

69
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So let's factor out (let me use a different color)

70
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-let's factor out a negative 6 over here. So

71
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plus negative 6 times (k plus 3). So now it looks

72
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like we can factor out a k plus 3; there's a k plus 3

73
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times a k, and then we have a k plus 3 times the

74
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negative 6. So let's factor that out- so we have this 8

75
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out front, that's not changing (let me write that and

76
00:04:19,432 --> 00:04:22,893
the brackets)- we're factoring out a k plus 3, so then

77
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we have a k plus 3 that we factored out, and then inside

78
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of that we just have left this k (instead of writing

79
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plus negative 6, I could just write k minus 6).

80
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We factor out the k+3 and we're done! And then we

81
00:04:42,608 --> 00:04:44,419
can rewrite this- this, the way we wrote it here

82
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is 8 times the product of k plus 3 times k minus 6,

83
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but we know from the properties of multiplication that

84
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this is the exact same thing as 8 times k plus 3

85
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times k minus 6. And we are done.