>> Here's a rule that will limit the number of possible trials that we make, either positively or negatively, it's called the upper and lower bounds rule and it works like this. Suppose we have this function and suppose we're investigating for zeroes, and we find f of zero to be 9 and we can do that by observation. If x is replaced with zero then we have simply 9. f of 1 again by observation we can determine that f of 1 is 2. Now, it kind of, it might look like to us that we have, let's see, zero goes to 9 and 1 goes to 2, so that graph is coming toward the x axis and so we were thinking well maybe we have a crossing here pretty soon. Let's try f of 2. Now, let's do that using synthetic division. So, we would go over here and put down the coefficients and the constant and bring down the 2, 2 times 2, 4 with negative 3 is 1. Two times 1 is 2 with negative 6 is negative 4. Two times negative 4 is negative 8, 9 minus 8, 1. Well, let's see 2 goes to 1. Alright, we're not quite there yet. Maybe we get there at 3, let's see. So, we bring down the 2, we have 3 times 2, 6, 6 minus 3, 3, 3 times 3, 9, 9 minus 3 -- excuse me 9 minus 6 is 3. Three times 3, 9, 9 and 9, 18, uh-oh, we're going in the wrong direction here. Now, do we continue to try? Do we try 4 and 5 and 6 and so forth? Well, gee we could analyze this and think well maybe there's a crossing, maybe there are a couple of crossings sort of in one of these intervals right in here to where we have 2 zeroes within a really tight interval. You see, integer value interval in here and so we might try this, this idea of the rational zero test or something like that. But the notion is that we could try positive integer values until we get really tired -- you know up to 10 and 12 and so forth and these numbers will get out of hand, but the upper and lower bounds rule will allow us to limit those numbers of trials right quick. Whenever we see that all of the signs here at the bottom of the synthetic division are positive, it tells us that this is an upper bound for the zeroes of the function. Upper bound for the zeroes of the function that is there are no real zeroes, there are no zeroes to the right of 3 that is greater than 3. Now, if zero occurs down here, zero can play the role of either a positive value or a negative value in this analysis process, so just you kind of bear than in mind. Now, just for informational purposes on the negative side there is a lower bounds rule as well. Suppose we're dividing on the negative side and we divide it by negative 2 and this just illustrates the sign pattern that needs to take place if we have a lower bound. And it's an alternating sign pattern, it's plus minus plus minus. And when we see that pattern on the negative side it means that this negative value is a lower bound and there are no real zeroes that are less than negative 2. So, we might as well stop the investigative process at that point.