WEBVTT
1
00:00:00.000 --> 00:00:12.000
Alright, what we're going to do is, for this problem, we need to find the, I forget,
2
00:00:12.000 --> 00:00:14.000
we need five to zero, that's what we need to do.
3
00:00:14.000 --> 00:00:19.000
And when we're finding the zeroes, we also need to determine what is going to be the multiplicity of the problem.
4
00:00:19.000 --> 00:00:21.000
I'm sorry, multiplicity of the zeroes.
5
00:00:21.000 --> 00:00:27.000
And we need to know what is the multiplicity, why are we figuring out what the multiplicity and who cares about multiplicity?
6
00:00:27.000 --> 00:00:30.000
Well, let's first start off finding the zeroes.
7
00:00:30.000 --> 00:00:33.000
Alright, and remember guys, we're talking about the zeroes.
8
00:00:33.000 --> 00:00:36.000
The zeroes are the same thing as the roots.
9
00:00:36.000 --> 00:00:39.000
You know, same thing as the x-intercepts.
10
00:00:39.000 --> 00:00:46.000
They all mean the exact same thing and what we're trying to do is, we're trying to find out when is our output value of our function zero.
11
00:00:46.000 --> 00:01:00.000
So we can say, finding the zeroes, we want to find the values of t that make g of t zero.
12
00:01:00.000 --> 00:01:08.000
Zero equals t to the fifth minus sixty cube plus nine t.
13
00:01:08.000 --> 00:01:15.000
Next thing, factor out your t, all these share a t, you can factor it out.
14
00:01:15.000 --> 00:01:25.000
Here's a six there.
15
00:01:25.000 --> 00:01:32.000
Stop, alert.
16
00:01:32.000 --> 00:01:38.000
We have something that's very, very special. Why we have a bigger parentheses than the smaller parentheses.
17
00:01:38.000 --> 00:01:42.000
The next thing that you guys see is we have a square number at the back.
18
00:01:42.000 --> 00:01:47.000
So automatically, whenever you guys see a square number, you can look for a perfect square.
19
00:01:47.000 --> 00:01:49.000
You can look for a difference of two squares.
20
00:01:49.000 --> 00:01:55.000
Always look for these things when you see a square number.
21
00:01:55.000 --> 00:02:00.000
And what we're having is we actually are going to be able to create a, you know, a perfect square.
22
00:02:00.000 --> 00:02:06.000
So what does that mean? Well, a square number means I'm going to take a number multiplied by itself, right?
23
00:02:06.000 --> 00:02:08.000
Three times three gives you nine.
24
00:02:08.000 --> 00:02:15.000
So does three and three give me six? Well, not positive three and three, but a negative three and three.
25
00:02:15.000 --> 00:02:23.000
So I can rewrite this as minus three times minus three.
26
00:02:23.000 --> 00:02:27.000
Oops, that's like way more.
27
00:02:27.000 --> 00:02:33.000
Now usually guys, you're used to this. Like everybody's so easy.
28
00:02:33.000 --> 00:02:39.000
Like you understand this? Oh, okay. You know, x minus three times x minus three.
29
00:02:39.000 --> 00:02:42.000
Right? That's easy. Everybody remembers, everybody's like, all right.
30
00:02:42.000 --> 00:02:44.000
I was going, I got it.
31
00:02:44.000 --> 00:02:46.000
Here, if you guys look at this, we're not dealing with x's name.
32
00:02:46.000 --> 00:02:50.000
We're dealing with T to the force and T squares.
33
00:02:50.000 --> 00:02:58.000
If you look at it, T squared times T squared is going to give you T to the fourth and the exact same thing.
34
00:02:58.000 --> 00:03:04.000
These T's when added up together are still going to be, or T squared is added up together are still going to be T squareds.
35
00:03:04.000 --> 00:03:09.000
So that's how I got the T squareds inside there.
36
00:03:09.000 --> 00:03:13.000
An awesome thing you didn't know now. Well, what do we do?
37
00:03:13.000 --> 00:03:19.000
T squared minus three times T squared minus three is the same thing as saying zero equals T times T squared minus three.
38
00:03:19.000 --> 00:03:31.000
And why is this so important? Well, because we're looking at multiplicity of a zero, we need to know when our zeros are written as factors.
39
00:03:31.000 --> 00:03:37.000
If they are to an even power, they are going to give us a multiplicity of two.
40
00:03:37.000 --> 00:03:41.000
And if they're to an odd power, they're going to give us a multiplicity of one.
41
00:03:41.000 --> 00:03:44.000
Now, we'll talk about multiplicity a little bit more in just a second.
42
00:03:44.000 --> 00:03:49.000
But for anyways, next thing we need to know is now we have a set of linear factors.
43
00:03:49.000 --> 00:03:55.000
I can say x times y equals zero. I know that one of these numbers has to equals zero.
44
00:03:55.000 --> 00:03:59.000
So we're going to set our two factors equals zero.
45
00:03:59.000 --> 00:04:05.000
T equals zero. And T squared minus three squared equals zero.
46
00:04:05.000 --> 00:04:12.000
Now, one real quick thing that a lot of students make mistakes on is they think that just because my variable is squared, it's a multiplicity of two.
47
00:04:12.000 --> 00:04:21.000
That is incorrect logic. What you need to do is not the variable power, but it's your factor if that's to raise to a power.
48
00:04:21.000 --> 00:04:25.000
So since this whole factor is raised to a power, it's a multiplicity of two.
49
00:04:25.000 --> 00:04:30.000
Just because my power is raised to two does not mean it's a multiplicity of two.
50
00:04:30.000 --> 00:04:33.000
So it's the factors not the powers.
51
00:04:33.000 --> 00:04:37.000
I'm sorry, it's the factors not just your individual variable.
52
00:04:37.000 --> 00:04:44.000
T squared equals zero is pretty easy and that's also to a multiplicity of one.
53
00:04:44.000 --> 00:04:52.000
Here, I got to square root to get rid of this square.
54
00:04:52.000 --> 00:04:59.000
There I am left with T squared minus three equals zero. Add three both sides.
55
00:04:59.000 --> 00:05:02.000
T squared equals three.
56
00:05:02.000 --> 00:05:11.000
Square root. T equals plus or minus the square root of three.
57
00:05:11.000 --> 00:05:16.000
Now, my factors, T is equal to zero.
58
00:05:16.000 --> 00:05:20.000
Square root of three. Negative square root of three.
59
00:05:20.000 --> 00:05:30.000
Both of these are to a multiplicity of two.
60
00:05:30.000 --> 00:05:35.000
So what it means is the graph is going to touch.
61
00:05:35.000 --> 00:05:38.000
Just touch. Just a little touch.
62
00:05:38.000 --> 00:05:47.000
It's going to touch at these two intercepts.
63
00:05:47.000 --> 00:05:55.000
And here, since as a multiplicity of one, it's going to intersect.
64
00:05:55.000 --> 00:06:00.000
So just remember, when you have a multiplicity of one, it's intersecting.
65
00:06:00.000 --> 00:06:04.000
In multiplicity of two, it's just going to touch.
66
00:06:04.000 --> 00:06:33.000
And that's how you determine what the zeros are and the multiplicity for you zeros.