Solving polynomial inequalities in factored form

0:00- [Narrator] So we're told that F of X is equal to all of this. Identify when F of X is greater than or equal to zero. So pause this video, have a go at that before we do this together. Okay, so what we're gonna do here is think about at what X values does F equal zero and then we can test the intervals in between because when it equals zero, it might just touch the X axis or it might cross the X axis.

0:27And then we can think about the intervals where it's actually greater than or equal to zero. So if we just set all of this equal to zero, so if we have negative three times x plus one times 2X minus three times X minus five equals zero, we know that if we take the product of a bunch of things and it equals zero, that means that any one of these things equaling zero would make this whole statement true.

0:52So we know that X plus one is equal to zero or it could, it could, or 2X minus three is equal to zero or X minus five is equal to zero. And so in this first one, you subtract one from both sides, you get X equal to negative one. Here you get 2X is equals three.

1:13When you add three to both sides, divide both sides by two. You get x equals 3/2. Over here you add five to both sides. You get X is equal to five. So these are the zeros. This is where our function is equal to zero. So let me just draw that on a bit of a number line.

1:29So there we go. And let's see. I have negative one, 3/2 and five. So let's say that this right over here is negative two. Negative, I need to make sure I have enough space. So negative one, this would be zero. Then I have one, two, three, four, five. And then let's throw a six in there as well.

1:58So our zeros are at X equals negative one, right over there, X equals 3/2, which is the same thing as one and a half right over there. And X equals five. So let's think about what is happening on every side or in between and on either ends of these zeros. So let's test what happens at F of negative two.

2:20So let me do that over here. Maybe we'll do it a different color. So F of negative two is equal two negative three. Negative two plus one is negative one, negative two. Let's see, two times negative two is going to be negative four minus three. Did I do that right?

2:41Negative four minus three is negative seven, negative seven. And then negative two minus five is negative seven again. So what is this going to be equal to? Well, we actually don't even have to think that much about what it actually equals to. The key is to just see whether it's positive or negative.

2:59And we have a negative times a negative, which is a positive, times a negative, which is a negative times a negative, which is going to be a positive. So this over here is positive. So another way to think about it is over this interval, we are greater than zero.

3:13So the function might look something like that. It actually looks like it's, it might be, it won't look exactly like this 'cause this is actually a fairly large number. So it probably looks something like this. But when we go less than negative one. Okay, now let's think about what happens in the interval between negative one and one and a half.

3:33So I will do that in orange. So let's just do F of zero. Zeros tend to be pretty easy things to calculate or when X equals zero. So that's gonna be negative three times zero plus one is one, two times zero is zero times negative three. So negative three and then zero minus five is negative five.

3:53So here we have, we have three negatives being multiplied. So this is going to be negative. You couldn't even see what it's going to be. It looks like it's like negative. Well it doesn't even matter. It's actually quite negative. So in this interval, we are getting quite negative, quite negative.

4:11I don't know exactly how far actually goes much further down. I haven't drawn any scale here or anything, but it's a negative over that interval. And now let's take the interval between one and a half and five and I will do that in maybe something purple. All right, so let's just take F of, I dunno, let's take F of three.

4:31So F of three. F of three is equal to negative three times three plus one, which is four times two times three, which is six minus three, which is three. And then three minus five is negative two. So four times three is positive 12. And the negative three times negative two is positive six.

4:53So this whole thing is going to be positive. So over this interval, we are going to be positive and you might say, wait, you just tested one number. How do you know you're positive over that whole interval? Well, if you weren't positive over that whole interval and we are assuming that this is a continuous function, that it doesn't have any jumps in it.

5:09And this is if you were to multiply this out, just a standard polynomial that would be continuous over all Xs. And so if it doesn't have any jumps, the only way it could turn negative is if it crossed the X axis again and then you would have another zero. But we don't have any zeros.

5:23These are the only zeros that we have. So that's why I feel good that it's positive in that interval. I don't know if it definitely looks like this, but it might look something like that depending on the scale. And then last but not least, let's think about when we are to the right of that last zero.

5:38So if I were to take F of, I already used the orange color. Let me do it in a different blue, F of, let me just pick F of six for example. That's going to be negative three times seven times two times six is 12 minus three is nine. So positive nine, and six minus five is one.

5:59So all of these are positive, but we have that negative three, so it's negative, so it's negative over that interval. So now let's think about and probably look something like this. It probably just goes negative, keeps going negative. But let's think about where to answer the question.

6:13Where does F of X greater than it equal to zero? Well, it's greater than or equal to zero for all Xs less than or equal to negative one 'cause it equals zero right there. So let me write this in a nice bold black here. So to answer the question, we get the Xs that are less than or equal to negative one, or we have this interval that is between 3/2 and five so, or 3/2 is less than or equal to X, which is less than or equal to five.

6:47These are the intervals over which F of X is greater than or equal to zero.