 In this video, we provide the solution to question number six for practice exam number three for math 1050. And we have to build a polynomial. It has real coefficients. It has degree three, which means it has three roots. And two of those roots are three and i plus one. Well, given that i plus one is a root, it's complex conjugate. It has to also be a root since there are real coefficients. And so that gives us the three roots that we're looking for. So then we have to essentially multiply out our polynomial at that moment. So we have x minus three right here. We have x minus one plus i. And then we have x minus one minus i. So I want to multiply out this part that can get a little bit t. So what I'm going to do is work it a little bit different. If I start off with x is equal to one plus i like so. What I'm going to do is I'm going to separate the real and imaginary parts so I get x minus one is equal to i. If you square both sides here, on the left-hand side, you're going to have to foil it out. So you get x squared minus 2x plus one. On the right-hand side, well, you have that i squared. That's a negative one. And adding one to both sides, you get x squared minus 2x plus two. That's equal to zero. So that gives us a polynomial who has a root one plus i, but it also has one minus i as a root in there as well. So I generally find that a simpler calculation than multiplying out these expressions right here. So this gives us that we have x minus three times x squared minus 2x plus two. Then we still have to multiply this thing out. No shortcuts on this one. So we distribute the x, you're going to get x cubed minus 2x squared plus 2x. Then distribute the negative three, you're going to get negative 3x squared plus 6x minus 6 like so. Combine like terms, we get x cubed minus 5x squared plus 8x minus 6. In which case, then that leads us to select choice B, which is the polynomial we just computed.