 Let's continue our discussion with the root and remainder theorems with something called the factor theorem. And the factor theorem centers around something called the root of a polynomial. So let capital X be some polynomial. If X equals A makes capital X equal to zero, then A is a root of our polynomial. And one way you can think about this is that a root of a polynomial X is a solution to the equation capital X equals zero. So what does this mean? Well, suppose X equals A is a root of some polynomial. Remember that a root is a solution to the equation X equals zero. So this means capital X equals zero when X equals A. Now our remainder theorem tells us that the value when X is equal to A is going to be the remainder when capital X is divided by X minus A. Since the value is zero by the remainder theorem, this means that when we divide by X minus A, the remainder is zero. But wait, there's more. Because we have our definition of quotient and remainder, if that remainder is zero, then we know that capital X is going to be X minus A times some other polynomial Y. And this suggests the first part of what's called the factor theorem. Suppose capital X is a polynomial. If X equals A is a root, then X minus A is a factor. But wait, there's more. Mathematicians like to turn things around. And so in this case, we know that if we have a root, then we have a factor. Well, suppose we have a factor. So suppose that X minus A is a factor of capital X. Well, then we know that capital X must be X minus A times something. So that tells us when lowercase X equals A, capital X must be equal to zero times something, otherwise known as zero. But remember, a root of a polynomial is a solution to the equation polynomial equals zero. So this means that X equals A is a root. And so this gives us the second half of the factor theorem. Suppose we have a polynomial. If X minus A is a factor, then X equals A is a root. Now in mathematics, a theorem is anything that has a proof. So we'll prove the factor theorem. Actually, we just did. Let's take on this a little bit more. Suppose capital X is a degree N polynomial. And furthermore, suppose that we know that X equals A is a solution to polynomial equals zero. Then remember, a root is a solution, which means that X equals A is a root. And our factor theorem says that if X equals A is a root, then X minus A is a factor. And so that means I can write my polynomial as X minus A times something, where the degree of our new polynomial is less than the degree of our original. Well, now let's consider that polynomial Y. Suppose we know that X equals B is a solution to capital Y equals zero. Well, then we know that since X equals B is a solution, we know that X equals B is a root. And by the factor theorem, if we have a root, we have a factor. So we know that Y is equal to X minus B times Z, which is some other polynomial where the degree of Z is less than the degree of Y. Remember, equals means that any time we see the one thing, we can replace it with the other. So this says any place we see a Y, we can replace it with X minus B times Z. Well, there's a Y here, so we can replace it. Well, now suppose we know that X equals C is a solution to capital Z equals zero. Then we know that the capital Z is X minus C times W, where the degree of W is less than the degree of Z. Equals means replaceable, so I can replace Z with X minus C times W, and now lather, rinse, repeat. The important thing to recognize here is that the degree of Y is less than the degree of X. The degree of Z is even lower. The degree of W is even lower and so on. And because our degrees keep dropping, we eventually arrive at some first degree polynomial, which we'll be able to write as X minus R times some zero degree polynomial, in other words, some constant. And what this means is that provided we can find these roots, then an nth degree polynomial can be written as a product of n linear factors. And this is something that's known as the fundamental theorem of algebra, version 1.0. Now, we typically make a big deal about this because this is a very important result. We call it the fundamental theorem of algebra, but the true significance of this result isn't clear until you hit calculus. And actually, it's not until you hit second semester calculus. And that's because the problem with this is that we need to know these solutions to these different polynomials. And if we don't know what these solutions are, it seems like we can't go any place. So how do we find these solutions? Well, we'll employ a very important strategy in mathematics, procrastination. In particular, we're not going to worry right now about how we actually find these roots, but instead, we'll assume that these roots exist and see where that takes us. We'll take a look at that next.