Analyzing positive and negative intervals of polynomials

So we have a function f of x that's written as the product of a bunch of first degree expressions. Now, if we obviously could also view this as a polynomial, especially if we expand this all out, it'll have our more traditional form. But what's nice about this form—and this is where we try to get when we're trying to find the zeros or the roots of a polynomial—is that it's very easy to pick out the zeros. The zeros here, these are the x values that make our function equal to zero.

Well, that's going to be x is equal to two, x is equal to negative 4. That would make this expression 0. x is equal to negative 5. Remember, if you make any one of these equal to 0, then the entire function is going to be equal to 0 because 0 times anything is 0. x equals 1, and x is equal to 9.

Now, as we've talked about in multiple videos, these are also going to be the—if these x values, when you input into the function, give you f of 2 equals 0 or f of negative 4 is equal to 0, then that means that they're also going to define our x-intercepts. What am I talking about?

So let me draw a big axis here. See, we go as low as x equals negative 5, as high as x equals 9. So let me draw something like this. So it's my y-axis, this is my x-axis. x is 1, 2, 3, 4, 5, 6, 7, 8, 9; 1, 2, 3, 4, 5.

All right, now let's just plot these. So when x is equal to 2, f of 2—we already know—is 0. If you input 2 here, then this expression is going to be zero. Zero times a bunch of stuff is going to be zero. So f of two is equal to zero, so the point (2, 0) is going to be on the graph of y equals f of x. So (2, 0) is that point right over there.

So notice x equals 2 is a zero and makes our function equal to 0 if you input it into it, and it's also the point (2, 0). It's going to be the x-coordinate of this point right over here. It's going to define our x-intercept.

Let's do it with the other ones. x is equal to negative 4. I can do that with a better color. x is equal to negative 4; negative one, negative two, negative three, negative four—right over there. x is equal to negative 5, x is equal to negative 5, x is equal to 1—and do that in pink. x is equal to 1, and x is equal to 9.

And x is equal to 9, that was that right over there—right. 1, 2, 3, 4, 5, 6, 7, 8.

Now, I'm not going to be able to exactly draw the graph of this function, but what's neat is I can realize, well, look, if these are the points where I'm going to intercept the x-axis, and these are the only points where I'm going to intercept the x-axis, the zeros—the real zeros—define all of the points at which I'm going to intercept the x-axis.

Because think about it—if there was a point that intercepted the x-axis that wasn't a zero, well, it wouldn't be a zero because the function would take on the value zero there. So between those points, the function will stay positive or will stay negative, and we can test that a little bit.

Let's think about what the function is doing between x equals negative 4 and x equals 1. So over this interval right over here, well, let's just pick a value there and then think about whether the function is going to be positive or negative at that value. An easy value to think about is maybe zero, and I just want to think about the sign—is the function going to be positive or negative when x is equal to 0?

So f of 0, f of 0—I'm just going to think about its sign. So the f of 0 is going to be 0 minus 2; well, that's going to be negative. 0 plus 4; that's going to be positive. 0 plus 5; that's going to be positive. 0 minus 1 is going to be negative. 0 minus 9 is going to be negative.

So we can see since at x equals zero we're going to get a—let's see—negative. This is going to be a negative times a negative, which would be a positive, and then we multiply by a negative again. So this is going to be equal to—this is going to get us to a negative.

So the function is going to take on a negative value here, and we know it's going to take on a negative value over this entire interval—over this entire interval—because if for some reason it took on a positive value over that entire interval, it's a continuous function, it would have to cross the x-axis, and that would define another zero. But we know there aren't any zeros between these two points.

So the function—I don't know what it looks like, but it's going to stay positive between those two values. Now as you go to the next interval, it's probably going to change signs, and we can verify that if you look at—if you look at say f of negative 4.5. So we'll do the same drill. f of negative 4.5, well, that's going to be—let's see—negative times a negative times a positive.

Negative 4.5 plus 5 is a positive; times the negative; times a negative—well, we have an even number of negatives being multiplied together here, so this is going to give us a positive. So this is going to give us a positive value. So we're going to have a positive value between these two points, and then we're likely to flip back to negative after that.

So then the graph is likely to flip back—flip back to negative right over there. You can test that. Try a really negative value. So f of negative 10, well, that's going to be a negative times a negative times a negative—let's see—times a negative times a negative times a negative.

So you have five negatives being multiplied together; you're going to have a negative. Hopefully, you see where all this is going. So this is—we're likely to switch signs here. You can test 1.5 and see if you get a positive value, then we're likely to switch back here.

Once again, the graph isn't going to look exactly like this. It might dip; it might be fairly shallow here, and then it might dip down a lot right here. I'm just making up the parts on the ups and downs, and then it probably does another sign change right over there.

We can verify that if f of positive 10—well, that's going to be a positive times a positive times a positive times a positive times a positive times a positive. So we're positive after that, and I don't know exactly what it looks like, but the key realization here is that the zeros are the only place that we're going to intercept the x-axis.

And between those zeros, if you're positive on that interval, you're going to be positive on any point. If you're positive at one point in that interval, you're going to be positive at any point in that interval. And if you're negative at any point in that interval, then you're going to be negative throughout the interval.

And of course, at the ends of the interval, you're going to be at 0. So if someone were to say, "What is the sign of f?" If someone were to say the sign of f on the interval on say negative 4 is less than x is less than 1—well, that's this interval; it goes from negative 4 to 1.

So that's the interval we actually started off with—that's that interval right over there. You'd say, "Oh, well look, f is always negative on this interval." If you don't include the endpoints—and we're not including the endpoints here; we don't have a less than or equal either point here, so here it's always negative.

If someone gave you a different interval, if they gave you an interval that includes where 0 is part of the interval—so someone said, for example, the interval from, I don't know, the interval from 1 to 9—well, that's going to include a 0 in it, so it's going to be positive.

It's going to be sometimes positive and sometimes negative on this interval. So the key is between the two roots, not including the two roots. If you have an interval between two roots, then you're always going to be positive or you're always going to be negative over that interval. Whatever the sign is at any point of the interval is going to sign throughout that interval—not including the endpoints.