WEBVTT
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That's part of our overview. Section 2.1 is the start of the panel message. We need
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to also talk about the multiplicity or the repeated factor of our panel elements. So,
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we're given, we talk about panel elements and we talk about zeros. And one way we're
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given the zeros, we can write it as a factor of x minus a. So, you know, a was our zero
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with your right is x minus a. Well, sometimes when you're finding factors, you're going
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to get a factor that's going to be raised to a power. Like, hence, when you factor, let's
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say it a factor on x squared. So, you're going to be getting a zero that's to that x squared
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power. Or here, you can say factor and you get x plus three squared. So, what does that
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mean? Well, there's a lot of really important parts that means. Depending on what your
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exponent is, if k is i, the graph crosses the x axis at x equals a. And two, if k is even
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the graph touches, but does not cross the y axis at x equals a. So, this is very important
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for us to be able to graph our graphs because we have a couple things. First of all, we need
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to know the beginning and end behavior. And that's from your leading coefficient test.
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Then the next thing is we need to find the zeros. And that's by, you know, understanding
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the definition of what a zero is and being able to factor. Then the third thing is understanding
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the multiplicity. And once we can kind of, you know, understand those three points, we
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really can get a good understanding of how to graph any part of no male function. And,
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you know, we'll get into a little bit more of, you know, using the intermediate value
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theorem and some other things. But, you know, we're given some easy polynomials. Let's
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take a look at this graph or this polynomial. This is the fact of polynomial. And so, let's
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take a look at their zeros. Now, remember, if here's my zero, I know that it crosses
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at a count of zero. That's going to be your x-intercept because that was one thing in the
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previous one that I told you we have. So, for here, I'm going to have x squared and
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then I have x minus 4 and x plus 2. So, first, I'm going to match up all my x-intercepts.
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I have zero, x equals 4 and negative 2. And then here I have x equals 4, 1, 2, 3, 4.
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And then I have a negative 3, 1, 2, 3. All right. So, a couple of things when looking
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at this one is, now, about knowing anything of our m behavior, or really dealing with that,
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I do know my m behavior is going to be greater than zero, all right? And, are my a's greater
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than zero because none of my x's are negative. And also, no, it's going to be the third,
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it's going to be to the fourth power. So, they're going to be going the same direction.
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And so, on this one, I know that my graph is going to end up my m behavior. It's going
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to look like this. All right. Now, since this is odd, I know it's going to go directly
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through those two points. Since my 2 is odd, it's going to cross it and since my 4 is
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odd, it's going to cross it. And then the one thing is someone's giving it touches,
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but it does not cross. So, that's going to look something like this. So, the graph,
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roughly, is going to take form of somewhere in the run in that shape. And let's say we
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took a look at this graph. Again, now, this one, again, I chose, you know, for simplicity,
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I chose my values of my x and of my leaving term be positive. So, it's going to be greater
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than zero. So, therefore, let's see, we got negative 4, so 1, 2, 3, 4. See, I know that's
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going to be, that's going to be going upwards. And then I'm negative 3. I know this is
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again going to open down because what happens is, I know that these are in my 2 end behaviors.
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This is just something I worked on from my first overview video if you watch it. You can
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determine how to find what the end behavior is. And since this is a multiplicity of 2,
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I know it's going to be open downwards. And then I can just connect the 2 graphs. And
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since this one is odd, I know it's going to cross. So, that's just a quick run over
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you of repeated zeros and how multiplicity works.