Positive and negative intervals of polynomials

- [Instructor] Let's say that we have the polynomial p of x. And when expressed in factored form, it is x plus two times two x minus three times x minus four. What we're going to do in this video is use our knowledge of the roots of this polynomial to think about intervals where this polynomial would be positive or negative. And the key realization is, is that the sign of a polynomial stays the same between consecutive zeros. Let me just draw an arbitrary graph of a polynomial here to make you appreciate why that is true. So x-axis, y-axis. And if I were to draw some arbitrary polynomial like that, you can see that between consecutive zeros the sign is the same. Between this zero and this zero, the polynomial is positive. Between this zero and this zero, the polynomial is negative. And that's almost intuitively true. Because if the sign did not stay the same, that means you would have to cross the x-axis. So you would have a zero, but we're saying between consecutive zeros. So between this zero and this zero, it is positive again. Then after that zero, it stays negative. Once again, the only way it wouldn't stay negative is if there were another zero. So now let's go back to this example here. And let me delete this because this is not the graph of p of x, which I have just written down. Let's first think about its zeros. So the zeros are the x-values that would either make x plus two equal zero, two x minus three equal zero, or x minus four equal zero. So first, we can think about, well, what x-values would make x plus two, x plus two equal to zero? Well, that, of course, would be x equals negative two. What x-values would make two x minus three equal zero? Two x minus three equal to zero, add three to both sides, you get two x equals three. Divide both sides by two, you get x equals 3/2. And then last but not least, what x-values would make x minus four equal to zero? (distant siren wailing) Add four to both sides, you get x is equal to four. And so if we were to plot this, it would look something like this. So this x equals negative two, (distant siren wailing) x equals negative one. This is zero. This is one, two, three, and four. And let me draw the y-axis here. So the y-axis would look something like this, x and y. We have a zero at x equals negative two, so our graph will intersect the x-axis there. We have a zero at x is equal to 3/2, which is 1 1/2, which is right over there. And we have a zero at x equals hour, which is right over there. And so we have several candidate intervals. And actually, let me write this down in a table. So the intervals over which and this is really, be between consecutive zeros, intervals to consider. So I'll draw a little table here. So you have x is less than negative two. That's one interval. X is less than, actually, let me color-code this. So if I were to say the interval for x is less than negative two, so that's this yellow that I draw at the extreme left there. We could have an interval where x is between negative two and 3/2, so negative two is less than x, is less than positive 3/2. That would be this interval right over here. You have the interval, I'm trying to use all my colors, between 3/2 and four, this interval here. So that would be 3/2 is less than x, is less than four. And then last but not least, you have the interval where x is greater than four, that interval right over there. So x is greater, greater than four. Now, there's a couple of ways of thinking about whether, over that interval, our function is positive or negative. One method is to just evaluate our p, our function at a point in the interval. And if it's positive, well, that's means that that whole interval is positive. It's negative, that means that that whole interval is negative. And once again, it's intuitive because, if for whatever reason, it were to switch, we would have another zero. I know I keep saying that. But another way to think about it is, over that interval, what is the behavior of x plus two, two x minus three, and x minus four? Think about whether they're positive or negative, and use our knowledge of multiplying positives and negatives together to figure out whether we're dealing with a positive or negative. So let's do it, we could do it both ways. So let's think of this as our sample x, sample x-value. And then let's see what we can intuit about or deduce about whether, over that interval, we are positive or negative. So for x is less than negative two, maybe an easy one or an obvious one to use, it could be any value where x is less than negative two, but let's try x is equal to negative three. So you could try to evaluate p of negative three. You could just evaluate that. Actually, let's just do that. So that's going to be equal to negative one times, two times negative three is negative six, minus three is negative nine, negative nine times negative three minus four, that is negative seven. So if you were to multiply all of this out, this would give you negative 63, which is clearly negative. So over this interval right over here, our polynomial is going to be negative. So then we can move on to the next one. An interesting thing is we didn't even have to figure out the 63 part. We can just see that there's a negative times a negative times a negative, which is going to be a negative. And so let's just do that going forward. Let's just think about whether each of these are going to be positive or negative and what would happen when you multiply those positive and negatives together. Now, in this second interval between negative two and 3/2, what is going to happen? Well, we could do a sample point. Let's say x is equal to zero. That might be pretty straightforward. Well, when x is equal to zero, we're going to be dealing with a positive times a negative times a negative, a positive times a negative times a negative. And the reason why I did that is in my head. I said, okay, that's going to be a positive two times a negative three times a negative four. So a positive times a negative times a negative, so I could write it this way. It's going to be a positive times a negative times a negative. Well, a negative times a negative is a positive, and a positive times a positive is a positive. So we are positive over that interval. And if you were to evaluate p of zero, you will get a positive value. Now what about this next interval? What about this next interval here between 3/2 and four? We could try x is equal to two. At when x is equal to two, we are going to get a positive times a positive times a, two minus four is negative, times a negative. So this is going to be negative over that interval. And then last but not least, when x is greater than four, we could try x is equal to let's say five. We are going to have a positive, positive times a positive times a positive. So we are going to have a positive. And as I mentioned, you could also do it without the sample points. You could say, okay, when x is greater than four, you could say, okay, for any x greater than four, if you add two to it, that for sure is going to be positive. For any x greater than four, if you multiply it by two and subtract three, well, that's still going to be positive 'cause two times something greater than four is definitely greater than three. And for any x greater than four, if you subtract four from it, you're still going to have a positive value. So that's another way to think about it, even if you don't use a sample point. But there you have it, we figured out the intervals over which the function is negative or positive. And we don't know exactly what the function looks like, but generally speaking it's negative over this first interval. So it might look something like this. It's positive over that next interval. And then it's negative over that third interval. And then it's positive over that last interval. So we'd have a general shape like this. We don't know, without trying out more points, exactly how high or low it would actually go.