 Hello, and welcome to a screencast about finding all the zeros of a polynomial. So I have a third degree polynomial here, f of x equals 3x to the third minus 14x squared minus 7x plus 10. And the way I always like to start off these problems is, you know, taking a look at our polynomial, making sure I have all my terms in there. So I've got a third degree here, a second degree here, a linear term here, and then my constant term here. Okay, so that's no problem at all, because if you're missing a term, sometimes I can throw things off. All right, next thing I want to do is go to my calculator. So let me put my calculator into the screen here. And in my y equals, I'm going to go ahead and put this polynomial. So 3x to the third minus 14x squared minus 7x. Sorry, my calculator is a little pokey here, and then plus 10. Okay, now if we're lucky, we will actually be able to, we do a zoom in number six for my standard window. If we're lucky, we can actually see some of these zeros. And sure enough, I see it's crossing my x-axis here one, two, three times. Okay, so we need to figure out what those three zeros are. Let's take a look at the table and see if we get lucky and have any of them occur at integers. Okay, so I see one down here at five zero, because a zero means that your y-value is zero. So I know that one of my zeros is going to be at x equals five. Okay, bring that calculator back up. If I, let's see, it looks like those numbers are going to be getting big. So let me scroll up and see, sure enough, at negative one, here's another zero. So I have one there, but I need three of them, because it's a third degree polynomial. And it looks like that one's getting really big in the negative direction. Oh, and they're just going to get worse. Okay, so it looks like there must be something here between zero and one, because my sign changes from positive to negative. Okay, so this is where we're going to want to use synthetic division to then help us keep knocking back this polynomial. Okay, so I'm going to set up my synthetic division with my zeros. So five is one of my zeros, and my coefficients are three, negative 14, negative seven, and positive 10. Okay, bring down your first number, and then we're going to be adding. So bring down our three. Now we're going to multiply, three by five is 15. We're going to add these, so we get a positive one. One times five is five, negative seven plus five is negative two. Negative two times five is negative 10, and that gives me a zero for remainder. Good, so that's kind of my gut check. I want to make sure that I don't have anything left over, because if I did, then five was not an actual zero. Okay, so that means my polynomial just got knocked back to 3x squared plus x minus two. Okay, so this is where you now have options. This is a quadratic. You can use a quadratic formula. You can try to factor it, but since we're practicing synthetic division, I'm going to do synthetic division again, so I'm going to use my other zero here of negative one, and I'm going to now just use this new polynomial here that I have for my quadratic. So my coefficients are going to be three, one, negative two. And if you're really savvy, you could actually put these two pieces of synthetic division together, but since I have limited screen room, I'm just going to do them side by side. Okay, again, we're going to bring our three down. Now let's multiply. Three by negative one is negative three, and then we're going to add one plus negative three is negative two. Multiply negative two by negative one, we get a positive two, remainder of zero. Okay, so that means that my last factor here is 3x minus two. But remember, we're trying to find our zeros, so if I set this equal to zero, I can go ahead and solve for x. So move our two over, 3x is two, divide by three, x is two-thirds. Okay, so my zeros are, let me list them out over here nicely, five, negative one, and two-thirds. Thank you for watching.