Lora Campeau: So we're going to connect some concepts from leading term analysis from section 3.5 to a factored polynomial, so that we can get an idea of what this polynomial looks like without much calculator work.
Lora Campeau: So the first thing I'd like to point out to you. This is a little bit different than our previous polynomials, I would like to point out that it's in factored form.
Lora Campeau: Our previous polynomials that we've been analyzing have been a summer difference of terms. So the way we would find our degree is slightly different.
Lora Campeau: And to avoid as many computations as possible. The idea is we do not want to have to multiply out all these different terms that would take us a long time. And then there's a more of a chance we make a minor error. So what I prefer to do if I am finding the degree
Lora Campeau: I will take the leading term in each of the factors. So what does that mean is, I would start off by. We do have our negative two. That's a Mano mule factor.
Lora Campeau: But then when I want to look at is the leading term in this binomial factor. So that leading term would be an x, I'll highlight there. So that'd be multiplied by just a positive x
Lora Campeau: Then I look at the leading term in the next
Lora Campeau: Binomial. And what happens is you want to pay attention to. If you were to expand this x plus one squared out, you would actually have x squared. So I'm going to take that to the power of two.
Lora Campeau: And taking the next leading term that's just a nice x will get that
Lora Campeau: And the next leading term if these had more X bonus, we would include the exponent on these values. So all we're doing is multiplying the leading terms of each of the factors together.
Lora Campeau: And then that's going to allow us to find whatever the first term was for our polynomial without having to expand it with all that foil and distribution.
Lora Campeau: So if I'm going to multiply this out and actually I should label this a little bit nicer. I am trying to find the degree but This my friends is really the leading term of the polynomial
Lora Campeau: So let's let's compute this.
Lora Campeau: We have negative two. Now what we're doing. Whenever you multiply with the same base you keep the base and add all the exponents. So we should have one plus two plus one plus one. So that ends up being of five.
Lora Campeau: So we now have a leading term to make judgments and kind of predict the behavior of our polynomial. So once we understand this, we can say okay
Lora Campeau: One important part, maybe would be to try to talk about endpoint behavior. So basically, now I can get my degree and my degree is the exponent which is a five. And that is an odd number. So that's telling me that the endpoint behavior is going to
Lora Campeau: Be different.
Lora Campeau: For the left and right edges of the graph. So remember we had an idea of they're either going to be what down up or up or down.
Lora Campeau: We narrow this down further by paying attention to the leading coefficient
Lora Campeau: Well, if the leading coefficient
Lora Campeau: Is equal to negative to its negative, this is going to further narrow down that we will have up down behavior for the endpoints. So I'll cross out the, down, up.
Lora Campeau: So we'll hold this off to the side until already to sketch. But this is a very important part of our polynomial
Lora Campeau: The next thing I'm going to do is deal with the roots and kind of some some concepts we've learned from this section is if your polynomial is in factor form you can quickly find the roots by setting each factor equal to zero. So the next thing I'll look for
Lora Campeau: I'll go with it this way. I'll kind of set up my roots.
Lora Campeau: I'm looking at x plus two equals zero, x plus one equals zero, x minus one equals zero and x minus three equals zero. So, right away.
Lora Campeau: It's going to give me my roots negative to negative one positive, one and positive three.
Lora Campeau: Now roots are also known as zeros are X intercepts. So right now I can draw a quick sketch of what's happening along the x axis.
Lora Campeau: So I'll take some time mark them off a little bit. I'll say each tick mark being worth one or so.
Lora Campeau: Trying to be as consistent as possible. And if I'm looking at plotting were negative two would be an X intercept, I have this right here.
Lora Campeau: Negative one would be over here. Positive one would be here and three would be here. So this is where my polynomial is crossing the x axis.
Lora Campeau: Now we can even deal with this a little bit better by identifying the multiplicity of each of the roots. So if I do this and maybe I might have to move my drawing over a little bit. I will do that.
Lora Campeau: For you all a little bit of time to move this down here a little bit. There we go.
Lora Campeau: Now I'm taking account for the multiple cities.
Lora Campeau: multiplicity of the roots change that back to the right color.
Lora Campeau: If you look at the corresponding factor for each of your roots, you look at the power that it's taken to so x plus two is taken to the power of one it's multiplicity is one
Lora Campeau: X plus one is taken to the power of to its multiplicity is to
Lora Campeau: X minus one is taken to the power of one it's multiplicity is one and x minus three is taken to the power of one it's multiplicity is one
Lora Campeau: Knowing whether your multiplicity is even or odd is now going to tell you what is happening at the root. So remember with odd multiplicity.
Lora Campeau: Looking at most of them.
Lora Campeau: One is an odd number. What that tells us is that the graph.
Lora Campeau: Is tangent, excuse me, not tangent it crosses I jumped ahead of myself.
Lora Campeau: So the graph crosses the x axis.
Lora Campeau: routes that have
Lora Campeau: And odd multiplicity. That's why it's so important to understand this.
Lora Campeau: So what I can do for myself for all these routes. So I'm looking at
Lora Campeau: Negative two, I have an understanding that in some case, it'll either cross going from down to up are up to down so I kind of write an X for this.
Lora Campeau: At x equals positive one, there's some sort of crossing happening. So I have x going on because I don't know which way it's coming from quite yet. I'll get there.
Lora Campeau: And it x equals three, there's some sort of crossing either it comes up to down or down the up and we'll go from figuring this out in a second.
Lora Campeau: The last route which is that negative one. I want to point out me see if I can
Lora Campeau: I will use a slightly different color running out of colors. I will use orange.
Lora Campeau: I want to pay attention at x equals negative one, you have an even multiplicity. So within even multiplicity. The graph is tangent
Lora Campeau: To the x axis.
Lora Campeau: At that route.
Lora Campeau: So that route corresponding route was negative one. So what I do to signify a tangent route. I know that it kind of
Lora Campeau: barely touches it in one place. So I'm kind of making a you. So it's either looking like a horse facing you are a downward facing you, I color coded so I don't get confused because it does kind of look like an x. But what this tells me
Lora Campeau: Is that now I kind of have an idea of the flow of my polynomial
Lora Campeau: So we are almost set to draw. We are going to combine this concept of multiplicity and routes with now leading term analysis.
Lora Campeau: So remember what I said about the up, down, part, we now have the direction of how our polynomial is supposed to look. So when we are seeing this, we can now say, Okay, if our polynomial is pointing upwards at the edges left edge.
Lora Campeau: Then it has to come from the top. So when I sketch my polynomial, it's coming from the top. And eventually, it has to cross the x axis at this negative to
Lora Campeau: Now the next thing I have to understand is, I have to account for the root of negative one. If the root of negative one is tangent it somehow has to come back up again and barely touch the x axis. So I'm using the the downward spacing you to understand this.
Lora Campeau: Now the flow of my polynomial continues in order for me to cross the x axis at positive one. My polynomial has to curve upwards again and cross through at positive one.
Lora Campeau: And in order for me to now cross the x axis at x equals three. My polynomial has to curve back downwards again. So it crosses and finally
Lora Campeau: At the end at points down. So we have this up and down behavior. And we have a pretty rough sketch of our polynomial
Lora Campeau: So this is a very nice way to analyze your function without having to pull out your Ti, I would like to point out that there's really no way at this point for us determine how
Lora Campeau: Low our local mins go or how well the tangent when we know
Lora Campeau: But we don't really know if you're not tangent to the X axis, your local mins in your local Max's could be a lot lower, or a lot higher than what you have in your rough sketch, but the main idea is to get a good idea of your function.
Lora Campeau: So I encourage you, if you want to take a look with me. Feel free if you feel good about this. You can stop the video at this point, but I would like to kind of show you how it compares to the actual graph. So one last thing I'm going to pull up the TI for us.
Lora Campeau: And
Lora Campeau: What I want to do is enter the polynomial in my function editor, so I'll remind myself. What we're actually looking at, I'm going to enter it in the factor form. No need to multiply anything out. So we have negative two.
Lora Campeau: gets multiplied by x plus two.
Lora Campeau: Which then gets multiplied by x plus one square x plus one.
Lora Campeau: Which then is taken to the power of two. So I'm just being real careful I'm clear on this.
Lora Campeau: We didn't have x minus one times x minus three.
Lora Campeau: Not net x minus 39 we have x minus three. There we go. So be careful on that I press Enter. Now,
Lora Campeau: What I'd like to do is understand when you made your drawings of your roots, you have a clear idea of how wide to make your X map. So let me zoom everything back to standard, just so that you could see what happens.
Lora Campeau: See, we have a pretty good idea of our, our X Men in our x max and really nice. In this sense, we can leave it as negative to positive 10
Lora Campeau: But you could kind of see it is sort of matching our function, we may have to. It looks like we are going to have to understand to see some Max's and mins
Lora Campeau: Looks like I'm gonna have to lower my why max and rate to me raise my why max and lower my Weidman so I'll do that with my window. Like I said, there's no quick way to do it. So sometimes I just go crazy to start at maybe negative 50
Lora Campeau: The idea is you can always shrink it down again.
Lora Campeau: So you graph it. And we're almost there. You'll notice that I think I have to raise my wine, Max, a little bit higher. I'll go with 100
Lora Campeau: And again, it's all experimentation and we have our function. Got a little crazy on the Y axis, but I like to kind of show you just a side by side idea. So I'll copy this.
Lora Campeau: If I can go back
Lora Campeau: To our actual graph.
Lora Campeau: And it's show you where we're dealing with and see if I can copy that that graph.
Lora Campeau: We'll see if that works. I'll paste
Lora Campeau: So look at this
Lora Campeau: So we can now kind of see
Lora Campeau: Our comparison of our graph graph will move it a little bit so you kind of look at a side by side, but check it out.
Lora Campeau: It looks like we're pretty darn accurate without needing a TI so super valuable again you'll see
Lora Campeau: That I already addressed this, but you kind of have when you're making these rough sketches. You'll notice that your, your mins and maxes if they're not change it to the x axis.
Lora Campeau: There's no way of knowing how high or high how low but see how how how nice that is just take a moment to admire that and appreciate the mathematics behind this without even needing to pull out the calculator.
Lora Campeau: Now I always encourage you if you're doing these problems. You can use your Ti to check. However, sometimes the degree is so high.
Lora Campeau: It ends up warping. The, the view screen of your Ti. So knowing these concepts and connecting 3.5 with 3.6 is definitely going to help you with a lot of these problems.
Lora Campeau: I hope this helps. I'll see you for the next example.
