 In this video I want to talk about the upper and lower bounds theorem. This is really two theorems to the price of one. You can of course have the upper bound theorem and you also have the lower bound theorem. Two tools that are very useful to help us factor large degree polynomials very quickly. The rational roots theorem gives us a potential list of rational roots and we can try them but we don't want to try every single one of them. The upper and lower bounds theorems can help us identify when we're making bad choices with respect to our potential rational roots. So if we have a f and this is some real valued polynomial, then when f of x is divided by x minus b, maybe we think b is a potential rational root, right? When f of x is divided by x minus b and I should mention that in this if to use the upper bound theorem b must be a positive number. It doesn't apply to negatives here. So if f of x is divided by a positive value x minus b and if you're using synthetic division then you have your three rows of synthetic division. The first one will be the coefficients, the bottom, the middle row has just has just the numbers that show up in the middle and then the bottom row will have the quotient and the remainder. Look at the bottom row. If the bottom row has no negative entries in it, no negatives whatsoever, that means that b is an upper bound for all the real roots of f. That means you picked a number that was too big. You need to try something smaller. If the numbers in the bottom row are all positive, you need to try something smaller. The lower bound theorem says that if f of x is divided by x minus a, which in this situation a is a negative number. If you're dividing by a negative number, you can use the lower bound theorem. It doesn't apply to positives. If you're using synthetic division and the bottom row turns out to be alternating signs and alternates between non-positive, non-negative, non-positive, non-negative switching signs, then that turns out your number is too small and you need to try something bigger. Let's try a specific example. Take f of x here to be x to the fourth minus three x squared plus two x minus five. If we do synthetic division here, write down the coefficients in descending order, you're going to get one, zero. Notice there's no x cubed term. We need a zero right there. Negative three, two, and negative five. Let's try dividing it by two, x minus two to be specific here. If you bring down the one, one times two is two, plus zero is two, times two is four, minus three is one, times two is two, plus two is four, times two is eight, minus five is three. You'll notice that the bottom row only contains positive. There's not a single negative there. What this tells us here is that we tried a number that was too big. If we want to find a positive root of the polynomial, we need to try something smaller. For example, we know for a fact that positive five will not work because if two was too big, then five is even worse. We should try something smaller. On the other hand, if we tried something like say, let's start this thing over again, one, zero, negative three, two, and negative five. Let's say we tried something like negative three. If you bring down the one, one times negative three is negative three, plus zero is negative three, times negative three is positive nine, minus three is a positive six, times negative three is going to be negative 18, plus two is a negative 16, times by negative three is going to give us a positive 48 minus five, you're going to get 43. You're going to notice here that the variation of signs, I change the signs, positive, negative, positive, negative, positive each and every time. Because it switches signs each and every time, this would tell us that negative three is too small. Now, I confess that for this polynomial, you should never try two or negative three because the rational roots test says we should be doing plus or minus one or plus or minus five. I admit that that's the case here. But you needless to say this still demonstrates the upper and lower bounds theorems right here. We had a number that was too big because there was no variation of signs right here. We had a number that was too small because we had a maximum variation of signs. Now, there's an interesting thing when it comes to the lower bound theorem that you're going to want to look out for here. If we take the polynomial g of x equals x to the fourth plus four x cubed plus three x squared plus seven x minus five. If we were trying to divide this one right here, notice the following. We write this down one, four, three, seven and negative five. I just wrote a plus sign, seven, negative five there. If we were to try and divide this by negative four, notice what happens in this situation. Bring down the one. One times negative four is negative four, plus four is zero. Times negative four would be zero, plus three is three, times negative four is negative 12, plus seven is a negative five, times negative four is positive 20, minus five would give us a 15, like so. And look at the signs right here. You're going to go from positive to zero to positive to negative to positive. So the nice thing about zero, zero is kind of like a wild card. Zero is this number that sits between the positive and negative. So it can be whatever you want it to be basically here. So in this case, zero is kind of like positive and negative. So I'd be like, oh, it's going to be a negative zero. So this is going to go from positive to negative to positive to negative positive. Because of the alternating signs right here, this would tell us that negative four is too small of a root. You need to try something bigger. So if you ever find a negative sign, it's going to be positive or negative, whatever makes the alternating thing worse. So negative zero would be okay in this situation. This is an example of something that's too small, and the lower bounds there would apply here.