 Welcome back to our lecture series math 1050, college algebra for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. In our previous two lectures in this series, lectures 26 and 27, we've talked a lot about factoring large degree polynomials. So we can find the real roots, which then help us solve polynomial equations. What I wanna do is continue in that vein, but in this lecture, I wanna talk about complex roots. So what if we wanna consider real roots and non real roots? So what if we allow for some imaginary component here? And this kind of helps complete a picture. And so this leads to what is commonly referred to as the fundamental theorem of algebra. Now, in mathematics, whenever we give a theorem a name, it's because we wanna reference it in the future. Like, so we can talk about theorem four, five, 22, right? But, you know, if I'm like, oh, oh, you remember four, theorem four, five, 22? No one's gonna remember what that number means, right? So we give students names so we can talk about them. And so you'll see things like, oh, the conjugate pair theorem, we can talk about that. So theorems often get names when we want to remember them. And so whenever a theorem is named in this series, you watching it should know it by name so we could reference it in the future. Now, whenever the adjective fundamental is added in front of the word theorem, this means it's a big deal. We don't add the term fundamental just loosely. There's very few theorems in mathematics that are deserving of the title fundamental. Fundamental theorem of algebra is one of these things. It's a pretty big deal here. And so what the fundamental theorem of algebra actually tells us is that every complex polynomial F has a degree N, which it's great equal to one, so it's not a constant polynomial. Every complex polynomial has at least a complex root. This is the thing is not every polynomial has a root. It depends on the context of numbers you're talking about. Like, if you take the polynomial X squared plus one, this polynomial has no integer root. It has no rational root. It has no real root. But it does have a complex root. Specifically, it's going to be plus or minus the square root of negative one, which we call I. And so the thing about so important about complex numbers is that there's never going to be a polynomial we can come up with which cannot be solved over the complex number system. Worst case scenario, we need to use real and imaginary numbers to describe the solution. That's what the fundamental theorem of algebra tells us. We never have to have a bigger number system than the complex numbers when we're working with polynomials. And then some consequences of the fundamental theorem of algebra. Every complex polynomial function F of degree N greater than or equal to one can be factored into N linear factors. And these factors don't necessarily have to be distinct. They could be repeated. So every polynomial will look like F of X is A to the N times X minus the first root times X minus the second root times X minus the third root all the way down to X minus the nth root. You always have this linear factorization. Now some of the factors could be repeated. And the roots R1, R2, these could be complex numbers, which be aware that real numbers are complex numbers. It's not an either or type of thing. It's not like, oh, a number is either real or imaginary. Real numbers are complex numbers, just like trees are plants, although not every plant is a tree. The complex numbers is a broader number system here. And so if you count the multiplicities of the roots, you will always have N roots when you're working over the complex number system. Now our goal is most to be focusing on real polynomials. We could talk about polynomials with complex coefficients. We're not really going to be interested in that. We're interested in real polynomials. But the roots of real polynomials could be complex numbers. But what we do know is that if a real polynomial, its coefficients are real numbers, if non-real roots show up, they're always going to come up and conjugate pairs. So if A plus Bi is a root of a real polynomial, then A minus Bi will also be a root. So once you find one complex root, you also get its conjugate. They always come as a package deal. So for example, suppose a polynomial f of degree five, whose coefficients are real, has roots one, five i, and one plus i. What are the two remaining roots? Well, so if we're building this polynomial f of x here, we don't have enough information to determine the leading coefficient. But since x is a root, since one is a root, excuse me, we get x minus one as a root. Because x minus, since five i is a root, you also get that, you're gonna get x minus five as a factor. But because five i is a root, that means its conjugate is also gonna be a root. The other conjugate is gonna be negative five i, which tells us that x plus five i is a root. And then if one plus i is a root, then that means x minus one plus i is a factor. But then the other root necessarily has to be its conjugate one minus i, which we're gonna get x minus one minus i as a root right here. And so this gives us the polynomial that is necessarily in play right here. And you might wonder, this is really a real value polynomial. Let's convince ourselves of that. If you take these conjugate pairs, foil them out. If you take x minus five i, and you times that by x plus five i, you'll notice when you foil, you're gonna get an x squared. You're gonna get a five i x minus five i x. And then you're gonna get a negative 25 i squared. Notice how this simplifies. You're gonna get five i x minus five x, they cancel each other. And remember that i squared is equal to negative one. So this turns out to be x squared plus 25. All right. And so that's pretty nifty right there. If you were to write f of x again, you're gonna get some coefficient a, we don't know what it is yet, x minus a, x minus one. Then you're gonna get x squared plus 25. Oh, those are real coefficients. What about the other part? What about the x minus one plus i and the x minus one minus i? We're gonna see the same thing happens if we foil those things out too. If we take x minus one plus i, and we take x minus one minus i, notice we are subtracting the roots. But we're only switching the signs here on the imaginary parts. If we multiply these things out, we're gonna get an x squared. We're going to get a minus one minus ix. We're gonna get a minus one plus ix. And then you're gonna get a positive one plus i times one minus i, like so. And so how do things combine this time? So you'll notice there's a negative ix and a positive ix, we're subtracting them. Those things are gonna cancel out, but the real part doesn't cancel out. You end up with an x squared. You're gonna get a negative one x and a negative one x. They actually combine to give you a negative two x. And what about these right here? If you multiply a complex number by its conjugate, you're always gonna get a sum of squares. This thing is gonna turn out to be one squared plus one squared, which is just a two. So that thing foils out to just to be x squared minus two x plus two. And so if we put that above here, x squared minus two x plus two, you'll now notice that this polynomial, it's not all the way multiplied out yet, but now it's a product of three real value, real value polynomials. If I were to continue with it, then we can see this is gonna be a real value polynomial. The complex conjugates basically interact with the such a way that the imaginary particle cancels out, and that's why they come in conjugate pairs. As another example, suppose we wanna find a polynomial of degree four whose coefficients are again real, and we want the roots to be one, one, and negative four plus i. So there's a repeated root. So what can we say about our polynomial? We don't have enough information for its leading coefficient, so we'll just say it's a for the moment. And because one is a repeated root, we're gonna get x minus one squared. We're gonna get x minus a negative four plus i. But then the other root has to be the conjugate, which is negative four minus i. So then the other one's gonna be x minus negative four minus i. And so we can foil it out to see what it is. If you don't wanna foil it out, I wanna mention another trick. If x equals negative four plus i, then let's do the following. This would mean that x plus four equals i. If we square both sides, well, you're gonna have to foil up the left-hand side. You're gonna get an x squared plus 8x plus 16. This is equal to negative one. And therefore, if you add one to both sides, you get x squared plus 8x plus 17. 17 equals zero. So if negative four plus i is a root, the smallest polynomial that has that as a root is this quadratic right here. That's gonna be the product of these two things right here. That kind of gives an alternative if you didn't wanna foil out all of this stuff right here. You can kind of avoid the arithmetic with complex numbers. And so our polynomial looks like a times x minus one squared times x squared plus 8x plus 17, which we could foil that, I think, out a little bit farther if we wanted to. I don't really care to do that right now. I mean, you can, of course, like x squared minus one, that would foil out to be x squared minus 2x plus one. And if you foil this by that, you end up with the following. Again, I'm just gonna put out the details. I'm not gonna do all of them, though. This would simplify to be x to the fourth plus 6x cubed plus 2x squared minus 26x plus 17. I'm just looking off my cheat sheet right now. I didn't do all of my head. But you can multiply that out and get that. But the point is at this stage, you could see that the coefficients were real. That the complex conjugates, the pairs, they're gonna cancel out their imaginary parts and you're gonna get real polynomials. So real polynomials will always come in these conjugate pairs. That's an important observation to make here.