 Let F be a field and let C be some number in that field. We call that number C a root, or some people like to call it a zero of a polynomial F of X, which I should mention F of X belongs to the polynomial ring F of join X, right? So C is a root of the polynomial F of X if the polynomial evaluated at the number C is equal to zero. This is also why people sometimes call these zeros of the polynomial. I'm personally not a big fan of that because I mean, zero is a number and that number actually could be a root. And so to say things like, oh, seven is a zero. It just seems a little bit confusing language. Don't get me wrong. In mathematics, we abuse notation and overload jargon all the time. But I feel like if it can be avoided, that can be done. And that's why I personally prefer the word root in this situation. Some people might call that a next intercept because we often think of F here as a function of real value function, maybe. F as a function from the real numbers to the real numbers. And so if you were to graph that function, the number C would in fact, then be an X intercept of the graph. But as we're not taking this analytic geometry approach, we're gonna avoid that terminology because after all F here is an arbitrary field. It could be a finite field for all we care. And this notion of a root still makes sense in that setting. So the roots of a polynomial are those numbers which evaluate the polynomial to be zero, okay? And so the main result from this video of lecture 20 that I want to present here is the so-called factor theorem, which actually tells us that there's a correspondence, a one-to-one correspondence between the factors of a polynomial and the roots of the polynomial because our interest in lecture 20 here is to explore factorizations of polynomials over an arbitrary field. In which case finding the roots coincides with finding linear factors. And therefore it's important to consider the search of roots of the polynomial as we try to factor polynomials. So let F be an arbitrary field and let little F be a polynomial whose coefficients come from this field. Then the factor, well, sorry, excuse me, that the polynomial X minus C is a factor of F of X if and only if F of C equals zero. So in other words, X minus C is a factor of a polynomial if and only if C is a root of the polynomial. So there is this one-to-one correspondence between linear factors and roots of polynomials. This is what I was alluding to just a moment ago. And this is gonna be a consequence of the remainder theorem. But we'll see that one just a second. So suppose that X minus C is a factor of F. Thus, there exists some polynomial Q such that F of X equals X minus C times Q of X. Now, I wanna be aware that this statement right here is not utilizing, this is not utilizing the division algorithm of any kind. It's just, hey, if you have a factor X minus C, this is what a factorization means. You could factor F to be X minus C times something. Now, if you evaluate this equation at the number C, well, the left-hand side is obviously F of C. The right-hand side, you're gonna get C minus C times Q of C. Don't care where Q of C is because C minus C is zero. Anything times zero is zero. So I want you to be aware what we have here. If X minus C is a factor, then C is a root. That will be true for any ring whatsoever. No comment is necessary about whether it's a field or a domain or anything. If you have a root of a polynomial, that produces a factorization of the polynomial, okay? Now, conversely, suppose we have a root. Now, in this setting, we're gonna use the remainder theorem. The remainder theorem told us, just as a reminder, the remainder theorem told us that if you divide a polynomial by the divisor X minus C, then the remainder in that division will then be the evaluation of the polynomial. Therefore, F of X, which will equal X minus C times Q of X plus F of C, right? So the remainder is the function evaluation in this setting. But since we're assuming that C is a root, the remainder will be zero, and therefore we have a factorization. So by the remainder theorem, we get the other direction as well, which the remainder theorem did actually use the fact that F is a field. Perhaps we could have had a somewhat weaker setting there, but as we get to Gauss's limit, which is the titular topic of this lecture, we'll see that how to relax this condition about a field. It doesn't necessarily have to be a field. A domain is actually a pretty good setting, and Bo will explain more about that, of course, in the next video. Now, before we close this video, I wanted to present a very important corollary of the factor theorem, and it's important enough that I'm actually not gonna call it corollary, I'm gonna call it a theorem, the so-called number of roots theorem. And so this is something you would have seen in like math 1050, college algebra, or your equivalent of what that would have been. This would have been done over like the real numbers or the complex numbers, but we can actually do this over any field this is just a consequence of the factor theorem which we just saw. So if you take a polynomial over the, if you take a polynomial whose coefficients come from some field F, then the number of roots of that polynomial is no more than the degree of the polynomial. Now, you can actually have different roots, right? So if you have something like X minus one times X minus two times X minus three, if this is your polynomial, let's say this is over the real numbers, right? Then clearly we can see that X equals one, two, and three are roots of the polynomial. This is a degree three polynomial, you can see when you multiply out the X's, you get X cubed there. This theorem is telling us that all of the roots are now accommodated, there's only three roots, there's not something else happening in this situation. But you don't have to have this maximal number of roots. For example, what if you took the polynomial X minus one squared, also viewed as a real polynomial, of course. Which of course we can multiply this out and get X squared minus two X plus one, something like that. It's degree two polynomial, but the only root is X equals one. This case has a repeated root, and so when you count the multiplicities, you get all of the roots, you get one and one, so you count it twice, but we're not gonna engage in this conversation about repeated roots yet. That's something we'll have to talk about in a future, in a future video because for rings, I should say fields of positive characteristic, this can get a little bit more confusing. The real numbers, of course, are characteristic zero. So again, I will postpone that conversation for a little bit later. And so let's take little f to be a polynomial in this ring, f a joint X, and let's assume it to be non-zero. Because if your polynomial f of X is equal to zero, if it's a zero polynomial, then you can evaluate at any number R and you end up with zero still. So every number in the field is a root of the zero polynomial, which is very likely that's gonna be a lot of numbers, right? But the zero polynomial, it doesn't have a degree, right? We did sometimes say, oh, the degree is negative infinity, but that was more of just a cave out, a cop out so that we could talk about formulas and such. Zero doesn't have a degree attached to it. When we talk about norms of domains, we don't define a norm for zero. And so zero, we don't have to talk about because even though it has every number as a root, it doesn't have a degree, so it's not a counter-example to this. So we do consider only non-zero polynomials. Let me first consider the situation where f is a constant polynomial, AKA a degree zero polynomial. Well, if f of X is some constants, it's gonna look like f of X equals C. If I evaluate that at some number R, you're still gonna get the C. And since this is not the zero polynomial, f of R is never equal to zero, it has no roots. And so that matches with the formula, okay? Now let's assume that the degree of f is a positive integer. So it's one, two, three, four, or something bigger, but it's gonna be a positive integer. And so now let's apply the factor theorem here. The factor theorem tells us that if R is a root of the polynomial, because maybe there's no roots, right? I mean, that seems kind of silly, but at the moment in the development of our theory, maybe there's no roots to the polynomial. I mean, after all, if you look at the real polynomial X squared plus one, as a real polynomial, it has no root. There is no real number whose square is equal to negative one. The complex numbers include a root, but the real numbers don't. And so this is gonna lead to this idea of extension fields that we will talk about later on in this lecture series, but in that situation, it's degree two, but the number of roots is zero, and zero is in fact less than two. It's in accordance with this theorem that we're talking about right here. So if the polynomial has a root R, that is F of R equals zero, then the factor theorem applies and we can factor it as F of X equals X minus R times Q of X here, which necessarily the degree of Q is gonna have the N minus one, okay, by the additivity of degrees. Now, if that was the only root, then we'd be fine, because remember, we're assuming N is a positive integer and therefore one is less than or equal to N. If R was the only root, we'd be done, we were great. What if there's another root, say S? Well, if we utilize this factorization, then F of S is gonna look like S minus R times Q of S, and that's equal to zero because S is a root right here. But looking at the factorization, S minus R times Q of S equals zero. Now, since F is an integral domain, that means F of joint X is also an integral domain. We've talked about this already. And so the only way a product of two quantities can be zero is if one of the factors is zero. Well, could it be the first factor? Well, R is not equal to S, which means S minus R is not equal to zero, so it's not that one. So the fact that F of joint X is a domain would then force it so that Q of S equals zero. But Q of S has strictly smaller degree than F of X here. And therefore, we can then invoke induction here. So the induction of hypothesis would be like, oh, it worked for everything smaller. So the inductive hypothesis would tell us that Q of X has less than, it has in minus one or fewer many roots. Well, if you would join the one root we had from before, that means that F of X will have in minus one plus one, here's a secret for everyone, that's equal to N, and that's then gonna be upper bound on the number of roots that one has. And so this is a pretty nice result here that over polynomials whose coefficients come from a field, the degree gives you an upper bound on the number of roots. And this is then a very nice corollary of the factor theorem.