 Our previous examples have shown us that a particular polynomial could have a lot of potential rational roots. The last example we saw in this lecture series had eight, the example before that had 12 possibilities. And so if we're just randomly guessing and checking potential rational roots, this could, we could potentially exhaust a lot of effort and not find these roots quickly. It could be the roots turned out to be the very last ones we were looking for, right? There's gotta be a way to sort of speed up this process. Now, we can't, this technique is not going to remove guessing and checking whatsoever, but we can make more educated guesses in the process. And there are some techniques to help us out here. The first one that we'll talk about in this video is referred to as Descartes' rule of variation of signs, named after the mathematician philosopher, Renee Descartes. Let F be a polynomial with real coefficients. Then the number of positive real roots of F right here, F of X, will either be equal to the number of variations in signs of F of X, or will be less than it by an even number. So what is that first principle mean right here? What do I mean by variation of signs? So if you take a polynomial right here, three X to the sixth plus four X to the fifth plus three X cubed minus X minus three. What I mean by variation of signs is that if you put the polynomial in descending order, you start with the biggest power and you go until the smallest power right here. Variation of signs means you go from coefficient to coefficient, does it switch? So you went from positive three to positive four, there's no variation of signs right there. As you go from four to three, there's no variation of signs that went from positive to positive. When you go from three to negative one, there is a variation of signs there. You went from positive to negative and therefore you count one variation of signs. And then you go from a negative to a negative, which in that case, there's no variation of signs here. So what we mean here by variation of signs is that our polynomial has a variation of signs of exactly one. And so what principle one right here tells us is that the number of positive real roots of f of x will be either equal to the variation of signs, which here is one, or it's less than it by an even number. So this tells us that the function f right here has, it will have exactly one positive real root. And so then that can help us know that as we're looking for real roots, we definitely have to find a positive one. And if we found a positive root, then we know not to look for any other ones. That can be a huge time saver. So what does it mean to subtract even number from it? Well, let's say that hypothetically, not for this polynomial, but let's say hypothetically, we had a variation of signs that turned out to be seven. Well, what Descartes rule says, the number of positive roots can be seven, or five, or three, or one. We just subtract two from it. So you look at all these odd numbers going down to one. Or if your variation of sign turned out to be six, you could get six, or four, or two, or zero. It could be there's none, actually. And so that's when we meet here. You subtract even numbers until you get to something negative. Negative is not a possibility here. That's how you can find the number of positive real roots. If you're interested in looking for negative real roots, the number of negative real roots of f of x is gonna be equal to the number of variations in sign to f of negative x, or less than that by an even whole number. So to find how many negative real roots you're gonna have, the idea is to look at f of negative x, which you're gonna replace each of the x's with a negative sign. So you get three negative x to the sixth, plus four negative x to the fifth, plus three negative x to the cubed, minus negative x minus three, like so. And then when you have an even power of a negative, it's gonna be a positive. You have an odd power of a negative, it's gonna be negative. So f of negative x is gonna become negative three x to the sixth. You're gonna get positive four x to the fifth minus... Oh, I'm sorry here. I did it backwards. With an even power right here, the negative's gonna go away. So you're gonna get three x to the sixth. Then with the odd one right here, you're gonna get negative four x to the fifth. This one here is an odd, so you get negative three x cubed. This one here is likewise odd, so you're gonna get positive x. It switches the sign to what was before, and you get negative three. And so now we wanna count the variations in this situation. So we go here, we have a positive to a negative, that's a variation of sign, so we count that. Then we go from a negative to a negative, no variation. We go from a negative to a positive, that's a variation of signs. And we go from a positive to a negative, that varies. So we're gonna get three with regard to the negative variation, which tells us that we're gonna have either, we're gonna have either three or one negative real root to this polynomial. That's what we can tell here. So this polynomial has either one real root or three or one negative roots. And so if I was to try to factor this polynomial, my temptation would be, let's try looking for negative roots. There's more negative roots, and so I'm more likely to find a negative root before I find a positive root. And once you find any root, whether it's positive or negative, that dramatically simplifies the problem and speeds up the process. Now, I do wanna mention, what about zero roots, right? Are we looking for positive? We talk about positive here, positive versus negative. How do you know if there's zeros of root or not? Well, one, you could check it, but the easier way is the only way that zero could be a root is if there's no constant term. Because if there's no constant term, that means everyone is divisible, but actually you could factor out the GCD and then X equals zero would be one of the roots right there. So finding zero's roots pretty easy. If there's a constant term, then zero's not a root. And so we don't have to put a lot of emphasis on that because we'll be able to detect it very easily.