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Polynomials - Example 6

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.


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Brieana R.

April 12, 2021

I am confused, for example 2) -2x^2--7x-4 aren't the zeros: -2,78078 and -0.719224 ???

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Okay, So for this polynomial, we want to do the following state. The degree described the in behavior and find all the zeros, if there are any. So what is the degree of this polynomial? Well, it's degree to because if I look at the highest power of X, look at the power. It's too now the in behavior. This is an even degree polynomial. So for your member from our chart, the leading coefficient is positive, which means that I can ignore the lower order terms. So if I just sketch what it looks like, well, I actually conduced sketch what this function is. It's a proble, but specifically because it's even. The degree is even, and the leading coefficient is positive. X is going as X goes to infinity at this, going to infinity and his ex goes to minus infinity. X is also F is also going to infinity. So there's thean behavior now to find the zeros we want Thio. Just use the quadratic formula. But remember, what is the quadratic formula say? It says negative B plus or minus square root of B squared minus four a c over to a and we looked at this discriminative. Have you said okay if this discriminate was negative, there no zeros. If it zero, there's 10 And if it's positive there are two zeros. But look at the discriminating here. First of all, let's note what a, B and C are so a is the coefficient of X squared. So a is one B is the coefficient of just X to the first, which is zero. And then see is the constant term, which is one so B squared minus four A. C is actually negative for which is less than zero. So there are no zeros for our functions because the discriminate is neck and we can actually see from the graph. I drew this accurately. This should pass through positive one here that the function never hits the x axis. The zeros are the points where the graph hits the X axis. Well, there aren't any. So there no zeros for this function. Okay, so, again, this is a degree to polynomial. So it's a quadratic in other words, agreed to because I look in the highest power of X is too Okay, so the in behavior this is again a quadratic. It's an even degree polynomial. So it's either going to be a smiley face or a frowning face, some sort of parabola. But because the leading coefficient here is negative, I know it's going to be inverted like that. So it's X goes to infinity. F is going to be going to minus infinity, and his ex goes to negative infinity. F is also also going to be going to negative infinity. Okay, So to find the zeros, I could use the quadratic formula, but I actually want to notice Negative two X squared minus seven X minus four. I'm looking for the X values that give me zero. I'm gonna notice that this will actually factor. And so how do I know it Factors? Well, it's just a little bit of practice. The more you see quadratic, the more you start to realize when they're going to factor in when they're not. And in this case, I've noticed that it factors. Now it's okay if you notice if you don't notice it, noticed that it factors because you could always just use the quadratic formula that will always work. But if you can see how it factors, it makes the problem a lot easier. And so I'll leave it to you to verify that if I foil this out, I'm gonna get exactly what I start. But now I know that either negative two x plus one is zero using the zero properties. Zero product property. It's a tankful. So the product of two, these two things have to be zero. So one of them has to be zero. So that's what I'm writing here. So either X is one half or excess for and I got one half by just at it. They're subtracting one and then dividing by Negative too. So here are our two zeros. One half and four. Okay, so this guy is a little bit more interesting. This is a degree for polynomial. So a Kordic and again I see that by noticing that the highest power of X is for this is again an even degree polynomial. So it's either going to be going to infinity is exposed to plus or minus infinity or minus infinity. And because the leading coefficient is positive, I know my function is gonna look something like this. No, it's not necessarily just going to be a parabola, but it's going to be coming down this way and then maybe doing some stuff and then going that way. Okay, so I don't really know what's going on. Kind of in the middle, but I know his ex goes to minus infinity. The function is approaching infinity. As is the case when X is going to plus infinity, the function is approaching infinity. Okay? And we'll actually learn how to draw what dysfunction actually looks like with more detail using some of the tools will develop in calculus, which is really cool. But for now, we just know the in behavior. We just know it's coming in from Affinity and then going off to infinity to the right. All right, so to find the zeros. Now, this is not a quadratic. We talked a lot about quadratic. So how do I find a zero here? Well, the reality is I really just need to guess. Okay. I know that sounds a little, you know, you know, shot in the dark, but sometimes you just have to guess. And now there are ways to figure that out. If you remember, there's something called the rational root there. Um and that's actually what I'm gonna use here to make an educated guess. I know that if there's a rational root of dysfunction, then if I look at this, Okay, the leading coefficient is one. The constant term is negative. Two. My options were basically divisor of to and just looking at this, I'm going to check Negative, too. So notice that if I just plug in negative too. What am I going to get? I'm going to get 16 minus 16 plus two minus two, which is zero. Okay, so I know that negative two is a zero. So how does that help me? Well, there's also something called the route factor serum that tells me if I can find a zero, then I can actually factor the function. I can write ffx as X plus two. Okay, so basically a factor that if I plug in negative two gives me zero a linear factor times some other pollen. Um, and what I need to do is actually figure out what this other polynomial is. And this other polynomial is going to be the quotient when I divide f by exposed to Okay. So what do I mean by that? I need to do some long division. I'm literally just going to divide X plus two into X to the fourth plus to execute. I'm gonna put plus zero x squared. Kind of like when you do division and yeah, 101. You leave a zero for the 10th spot and then minus one x minus two. Okay, so how does this work? Well, X Plus two goes into X to the fourth execute times. And so then I can multiply this out. I have X to the fourth plus to execute and then nothing else and then also tracked. So I'll just get Syria. But then I need to bring down the zero X squared the minus x the minus two. So I just have minus X minus two and X plus two goes into negative X minus two minus one times. So then I multiply and get minus X. It's too. I did zero. Okay, so my remainder is zero. My question is, execute minus one. So what I've done is I've factored my polynomial as exports to times execute minus one. You can verify that if you multiply oil this out, you'll get back where you started. But Look at this. Execute minus one. This is a difference of cubes. And what I mean by that is it's X cubed minus one cube. So it actually factors as X minus one times x squared plus X plus one. Okay, so now I'm really making some real progress. I have two linear factors and then this quadratic and actually the quadratic factor. If I look a b squared minus four a c, the discriminate, it's gonna be negative is gonna be one. So B square was one minus four. That's negative. Three, which is less than zero. So that means this quadratic factor has no zeros. So if I set this equal to zero, if I set f of X equal to zero, this factor is never zero. So one of these two factors has to be zero. So x plus two equals zero or X, minus 10 So then I have two zeros to this function. X equals negative too, and X equals one. And there's a be the only heroes of this court IQ Parliament