 Hello, and welcome to another screencast about finding zeros of a polynomial. But this time we're gonna be looking at a quartic, a fourth degree polynomial. So I have a fourth degree, I've got a third degree, second degree, my linear term and my constant. Okay, so again we're not missing anything, so that's good. All right, well just like we started the last one, let's go to the calculator first. So let me bring that up and type this polynomial into my y equals. So I've got x to the fourth minus 2x to the third plus x to the second minus 8x minus 12. Okay, now this is a fourth degree polynomial, so we are expecting four zeros. Let's take a look at the graph. Oh, I only see two. Okay, so that means two of them are gonna be real, two of them are gonna be imaginary. Okay, this is why we looked at imaginary numbers before. Okay, take a look at the table again and let's see if we get anything off of there, and we sure do. I see a zero here at negative one, because my y value is zero. And let me toggle around and see where my other one was at. Cuz it looked from the graph, there was one positive, one negative. And sure enough, there's a positive one here at three. Okay, so right now my real zeros are at x equals negative one and x equals three. And these came from our table. Okay, again, just like we did in the last one, let's go ahead and start knocking back this polynomial. That's a fourth degree, we're gonna be able to knock it back twice. So that'll knock it back to a second degree. And then we can use our quadratic formula or maybe solve it from there, we'll have to see what we get. All right, so I'm gonna start with synthetic division in negative one. It doesn't matter which one I start with. I'll just start with the one I listed first, so I don't forget. And then I'm gonna list out my coefficients. So that'll be one, negative two, one, negative eight, and negative 12. Okay, and again, our gut check when we do this synthetic division is we better end up with a remainder down here of zero. Okay, drop down my first term, multiply. So one times negative one is negative one. Add these, I get negative three. Multiply, I get positive three. Add, I get positive four. Multiply, I get negative four. Add, I get negative 12. Multiply, I get positive 12. And sure enough, our remainder is zero. Okay, that's good. So I've now taken my fourth degree and I've knocked it back to a third degree, okay? So I'm just gonna make a note here. This is a third degree, because I'm now gonna go ahead and take my other zero, my three, and I'm just gonna go ahead and do synthetic division right off of these other numbers we got. Okay, so again, I'm gonna drop this one down. So drop our one down, multiply. So three times three is three. Add, we get zero. Multiply, zero times three is zero. Add, we get four. Four times three is 12. Add, we get zero, and that is our remainder. Okay, so I had a third degree and I now brought it down to a second degree. Again, our gut check is we got a zero, so that's good for our remainder. So this is now a second degree polynomial. So that means this one is my x squared term. The zero is the coefficient of my x, oopsie, x I said. And this then is my constant. So that means I now have to solve the polynomial x squared plus four equals zero, okay? Well, I can go ahead and add my, I'm sorry, subtract my four from both sides, what was I thinking? So x squared equals negative four. Do the square root of both sides. So x equals plus or minus, don't forget about that part. Square root of negative four, and sure enough, these are gonna be imaginary. So x is plus or minus two i, okay? So it's good that these come in pairs, because remember they need to come as conjugates. So that way when you go to multiply things out, you end up getting something that has real coefficients. Okay, so my final answers are negative one, three, two i, and negative two i. All right, thank you for watching.