1
00:00:00,000 --> 00:00:00,510


2
00:00:00,510 --> 00:00:02,399
We're asked to factor this expression.

3
00:00:02,399 --> 00:00:05,459
And there's going to be simpler ways to factor it, but

4
00:00:05,459 --> 00:00:08,390
in this video I'm going to factor it by grouping.

5
00:00:08,390 --> 00:00:10,460
And when you factor by grouping, what you need to do

6
00:00:10,460 --> 00:00:13,170
is think about two numbers whose products are

7
00:00:13,169 --> 00:00:14,050
going to be equal to.

8
00:00:14,050 --> 00:00:16,530
You have actually one coefficient right here, right?

9
00:00:16,530 --> 00:00:19,020
t squared is the same thing as 1t squared.

10
00:00:19,019 --> 00:00:21,070
So we're looking for two numbers, let's call them a and

11
00:00:21,070 --> 00:00:23,510
b, a times b.

12
00:00:23,510 --> 00:00:25,750
the product of these two numbers needs to be the

13
00:00:25,750 --> 00:00:29,820
product of the coefficient on the t squared, which is 1, and

14
00:00:29,820 --> 00:00:32,090
the negative 15 right here.

15
00:00:32,090 --> 00:00:35,950
So a times b has to be equal to 1 times negative 15, or

16
00:00:35,950 --> 00:00:38,510
just negative 15.

17
00:00:38,509 --> 00:00:43,769
And the sum of a and b, and a plus b, needs to be equal to

18
00:00:43,770 --> 00:00:45,035
negative 2.

19
00:00:45,034 --> 00:00:49,609


20
00:00:49,609 --> 00:00:51,939
And once we have these two numbers, I can show you how we

21
00:00:51,939 --> 00:00:54,239
can use those to factor by grouping.

22
00:00:54,240 --> 00:00:56,620
And in other videos I've actually broken down to why

23
00:00:56,619 --> 00:00:58,429
this technique works.

24
00:00:58,429 --> 00:01:01,369
Now, let's think of the different factors of negative

25
00:01:01,369 --> 00:01:04,289
15 when we take their product, and if we take the sum, if we

26
00:01:04,290 --> 00:01:06,030
can somehow get to negative 2.

27
00:01:06,030 --> 00:01:10,590
So let's look at the different factors of negative 15.

28
00:01:10,590 --> 00:01:15,260
So we could do-- let me do it in this other, let me do it in

29
00:01:15,260 --> 00:01:19,250
pink-- see if 1 and negative 15, these are-- so everything

30
00:01:19,250 --> 00:01:22,569
I list here, their product is going to be negative 15.

31
00:01:22,569 --> 00:01:24,219
But let's think about what happens when

32
00:01:24,219 --> 00:01:25,789
you take their sum.

33
00:01:25,790 --> 00:01:28,790
So 1 and negative 15, the sum is negative 14.

34
00:01:28,790 --> 00:01:33,120
And if you did negative 1 and 15, you're just going to get

35
00:01:33,120 --> 00:01:33,930
the negative of that.

36
00:01:33,930 --> 00:01:34,730
You're going to get 14.

37
00:01:34,730 --> 00:01:36,240
It does not equal negative 2.

38
00:01:36,239 --> 00:01:42,089
So what happens if you take 3 and negative 5?

39
00:01:42,090 --> 00:01:45,430
So their product is definitely negative 15, 3, plus negative

40
00:01:45,430 --> 00:01:48,150
5 is negative 2, so that works.

41
00:01:48,150 --> 00:01:51,910
And if we tried negative 3 and 5 first, we would have gotten

42
00:01:51,909 --> 00:01:52,780
that to be positive 2.

43
00:01:52,780 --> 00:01:54,466
It's just like, oh, we just have to swap the signs and we

44
00:01:54,465 --> 00:01:55,509
would have gotten a negative 2.

45
00:01:55,510 --> 00:01:58,280
So these work, 3 and negative 5 work.

46
00:01:58,280 --> 00:02:02,430
3 times negative 5 is negative 15, 3 plus negative 5 is

47
00:02:02,430 --> 00:02:03,420
negative 2.

48
00:02:03,420 --> 00:02:06,129
So what we want to do here is break this

49
00:02:06,129 --> 00:02:07,229
middle term up here.

50
00:02:07,230 --> 00:02:10,550
We know that 3 plus negative 5 is equal to negative 2.

51
00:02:10,550 --> 00:02:14,010
So we can break up this middle term here as a sum of-- and

52
00:02:14,009 --> 00:02:15,370
I'll do it right here; I'll actually do it in the same

53
00:02:15,370 --> 00:02:18,909
color-- so this thing here, we can rewrite as t squared.

54
00:02:18,909 --> 00:02:21,229
I'll put the minus 15 out here.

55
00:02:21,229 --> 00:02:27,319
But the negative 2t we can rewrite as the sum of 3t.

56
00:02:27,319 --> 00:02:34,959
We could write it here as plus 3t, minus 5t.

57
00:02:34,960 --> 00:02:38,520


58
00:02:38,520 --> 00:02:41,070
And when you're trying to figure out which one to put

59
00:02:41,069 --> 00:02:44,539
first or second, you should look at these other terms, and

60
00:02:44,539 --> 00:02:46,289
say which ones have common factors?

61
00:02:46,289 --> 00:02:51,569
The 3 and 5 both have a common factor with 15, so it's not as

62
00:02:51,569 --> 00:02:53,459
obvious which one to put first, so we're just going to

63
00:02:53,460 --> 00:02:53,890
go with this.

64
00:02:53,889 --> 00:02:58,250
3t minus 5t, I got that from 3t minus 5t is equal to

65
00:02:58,250 --> 00:02:59,389
negative 2t.

66
00:02:59,389 --> 00:03:02,299
Positive 3 times negative 5 is equal to negative 15.

67
00:03:02,300 --> 00:03:03,689
That's where it came from.

68
00:03:03,689 --> 00:03:06,479
Now, we're ready to factor by grouping.

69
00:03:06,479 --> 00:03:09,530
So let's take the first group, let's take these first two

70
00:03:09,530 --> 00:03:11,229
terms, right there.

71
00:03:11,229 --> 00:03:13,060
And what's the common factor there?

72
00:03:13,060 --> 00:03:14,509
Well, the common factor there is t.

73
00:03:14,509 --> 00:03:17,810
So if I factor a t out, that becomes t times-- t squared

74
00:03:17,810 --> 00:03:19,270
divided by t is t.

75
00:03:19,270 --> 00:03:22,530
3t divided by t is 3.

76
00:03:22,530 --> 00:03:25,120
So these first two terms are the same thing as t

77
00:03:25,120 --> 00:03:27,090
times t plus 3.

78
00:03:27,090 --> 00:03:32,500
Now, let's look at the second two terms.

79
00:03:32,500 --> 00:03:34,150
What's a common factor?

80
00:03:34,150 --> 00:03:36,680
Well, they're both divisible by negative 5, so let's factor

81
00:03:36,680 --> 00:03:37,930
out a negative 5.

82
00:03:37,930 --> 00:03:40,680


83
00:03:40,680 --> 00:03:44,450
And negative 5t divided by negative 5, if you factor out

84
00:03:44,449 --> 00:03:46,399
the negative 5, you're just going to have a t there.

85
00:03:46,400 --> 00:03:49,620
And then negative 15, if you factor out a negative 5, you

86
00:03:49,620 --> 00:03:51,700
divide negative 15 by negative 5, you're just going to have a

87
00:03:51,699 --> 00:03:54,969
positive 3.

88
00:03:54,969 --> 00:03:59,289
And then notice, you now have two terms here, two products,

89
00:03:59,289 --> 00:04:05,659
and they both have the common factor of t plus 3.

90
00:04:05,659 --> 00:04:10,199
So we can rewrite this right here as a product of t plus 3.

91
00:04:10,199 --> 00:04:12,564
We're undistributing the t plus 3.

92
00:04:12,564 --> 00:04:20,139
We're factoring out the t plus 3. t plus 3 times t, right?

93
00:04:20,139 --> 00:04:23,919
Times t minus 5.

94
00:04:23,920 --> 00:04:26,310
And I want you to really make sure you feel good that these

95
00:04:26,310 --> 00:04:28,030
are really the same thing.

96
00:04:28,029 --> 00:04:31,404
If you take t times t plus 3 and factor out the t plus 3,

97
00:04:31,404 --> 00:04:33,549
you're just left with that t.

98
00:04:33,550 --> 00:04:36,180
If you take negative 5 times t plus 3 and you factor out the

99
00:04:36,180 --> 00:04:38,949
t plus 3, you're just left with that negative 5.

100
00:04:38,949 --> 00:04:41,255
But once you factor out the t plus 3, and you're just left

101
00:04:41,255 --> 00:04:44,469
with the t minus 5, you have fully factored

102
00:04:44,470 --> 00:04:45,830
this expression here.

103
00:04:45,829 --> 00:04:47,579
And in the future we're going to see easier ways of doing

104
00:04:47,579 --> 00:04:50,370
this, but factoring by grouping is actually the

105
00:04:50,370 --> 00:04:55,129
easiest way to do it if you have a coefficient higher than

106
00:04:55,129 --> 00:04:56,389
1, or a non-one coefficient.

107
00:04:56,389 --> 00:04:58,829
It could also be a negative coefficient out front here.

108
00:04:58,829 --> 00:05:00,649
When you have 1 as your coefficient here, there's

109
00:05:00,649 --> 00:05:03,709
actually much easier ways to factor something like this,

110
00:05:03,709 --> 00:05:06,120
but it's really the same thought process.

111
00:05:06,120 --> 00:05:06,800