1
00:00:00,998 --> 00:00:04,110
Factor t squared plus 8t plus 15.

2
00:00:04,110 --> 00:00:07,268
So let's just think about what happen if we multiply 2 binomials

3
00:00:07,268 --> 00:00:10,751
t plus a times t plus b

4
00:00:10,751 --> 00:00:13,769
And I'm using t here because t is the variable in the polynomial

5
00:00:13,769 --> 00:00:16,556
that we need to factor. So if you multiply this out,

6
00:00:16,556 --> 00:00:20,225
applying de distributive property twice or using F.O.I.L,

7
00:00:20,225 --> 00:00:25,519
You get t times t which is t squared plus t times b

8
00:00:25,519 --> 00:00:31,742
which is bt plus a times t which is at plus a times b

9
00:00:31,742 --> 00:00:35,085
which is ab. We essentialy multiplied every term here by

10
00:00:35,085 --> 00:00:39,358
every term over there. And then we have 2 terms, 2 t terms

11
00:00:39,358 --> 00:00:41,726
I guess we could call them. This bt plus at.

12
00:00:41,726 --> 00:00:44,559
So we can combine those. So we get t squared

13
00:00:44,559 --> 00:00:49,342
plus a plus b t, I can write this is b plus a t as well.

14
00:00:49,342 --> 00:00:55,380
plus ab. If we compare this to this right over here.

15
00:00:55,380 --> 00:00:57,191
we see that we have a similar pattern.

16
00:00:57,191 --> 00:01:00,860
Our coeffient, Our coeffient on the second degree term is one

17
00:01:00,860 --> 00:01:04,110
Our coeffient on the second degree term here is one.

18
00:01:04,110 --> 00:01:07,918
We didn't have to write it. Then, a plus b is a coefficient on t.

19
00:01:07,918 --> 00:01:12,330
So 8, right over here, this 8 could be a plus b.

20
00:01:12,330 --> 00:01:16,835
And then finally, our constant term ab that could be 15.

21
00:01:16,835 --> 00:01:20,689
That could be 15. So if we want to factor this out.

22
00:01:20,689 --> 00:01:24,637
We just have to find an a and a b where their product is 15

23
00:01:24,637 --> 00:01:29,374
and their sum is 8. In general, in general if you ever see

24
00:01:29,374 --> 00:01:32,253
I'll write it in kind of the more traditional with the x variable

25
00:01:32,253 --> 00:01:38,151
If you see anything of the form x squared plus bx plus c.

26
00:01:38,151 --> 00:01:42,237
The coefficient here is 1. Then you just have to find

27
00:01:42,237 --> 00:01:46,835
2 numbers whose sum is equal to this thing right here

28
00:01:46,885 --> 00:01:49,946
and his whose product is equal that thing right there.

29
00:01:49,946 --> 00:01:54,358
Whose sum is equal to 8 and whose product is equal to 15

30
00:01:54,358 --> 00:01:56,820
So what are 2 numbers that add up to 8 and

31
00:01:56,820 --> 00:01:59,699
whose product are 15. So if we just factor 15,

32
00:01:59,699 --> 00:02:02,950
We have 1 and 15. Those don't add up to 8 in anyway.

33
00:02:02,950 --> 00:02:06,154
3 and 5. Those do add up to 8.

34
00:02:06,154 --> 00:02:10,519
So a and b could be 3 and 5, So a and b,

35
00:02:10,519 --> 00:02:15,070
this could be 3 times 5 and then 8 is 3 plus 5.

36
00:02:15,070 --> 00:02:17,903
Now we can just go straight and factor this and say, hey,

37
00:02:17,903 --> 00:02:22,362
This is t plus 3 time t plus 5. Since we already figured out what a and b are.

38
00:02:22,362 --> 00:02:26,495
But what I want to do is kind of factor this by grouping.

39
00:02:26,495 --> 00:02:29,745
So, I'm essentially going to go in reverse step.

40
00:02:29,745 --> 00:02:33,321
From what I just showed you. So this first, this polynomial right here,

41
00:02:33,321 --> 00:02:37,362
I'm going to write it as t squared plus, instead of

42
00:02:37,362 --> 00:02:41,355
8t i'm going to write it as a sum of at plus bt.

43
00:02:41,355 --> 00:02:47,486
Or as a sum of 3 t plus 5 t. So plus 3t plus 5t.

44
00:02:47,486 --> 00:02:51,061
So, I'm essentially, I starded here and I'm going to this step

45
00:02:51,061 --> 00:02:55,659
where I break up that middle term into the coefficients

46
00:02:55,659 --> 00:02:58,863
that add up to the 8. And then finally plus 15.

47
00:02:58,863 --> 00:03:01,232
and now, I can factor by grouping.

48
00:03:01,232 --> 00:03:04,483
These 2 guys right over here have a common factor t.

49
00:03:04,483 --> 00:03:07,919
And these 2 guys over here have a common factor 5.

50
00:03:07,919 --> 00:03:11,495
So let's factor out the t in this first expression over here

51
00:03:11,495 --> 00:03:16,418
or this part of the expression. So it's t times t plus 3 plus

52
00:03:16,418 --> 00:03:19,761
and then over here if you factor out a 5, you get a 5.

53
00:03:19,761 --> 00:03:27,145
times t plus 3, 5t divided by 5 is t, 15 divided by 5 is 3.

54
00:03:27,145 --> 00:03:29,746
And now you can factor out a t plus 3.

55
00:03:29,746 --> 00:03:33,972
You have a t plus 3 being multplying times both of these terms

56
00:03:33,972 --> 00:03:36,619
So, let's factor, let's factor that out.

57
00:03:36,619 --> 00:03:43,121
So it becomes t plus 3 times, times, t plus 5

58
00:03:43,121 --> 00:03:46,278
times, I'll write that plus a little bit neater,

59
00:03:46,278 --> 00:03:54,405
times t plus 5. And we are done. We didn't actually

60
00:03:54,405 --> 00:03:56,913
have to do this grouping step altough hopefully you see

61
00:03:56,913 --> 00:03:58,910
that it does work. We could have just said look

62
00:03:58,910 --> 00:04:01,789
from this pattern over here I have two numbers

63
00:04:01,789 --> 00:04:04,158
that add up to 8 and their product is 15.

64
00:04:04,158 --> 99:59:59,999
So, this is t plus 3 times t plus 5. or t plus 5 times t plus 3 either way