 I want to look at just one more example in which we apply the fundamental theorem of algebra and also the conjugate zeroes theorem. So if you look at your handout, the conjugate zeroes theorem says if a polynomial f of x has only real coefficients and a plus bi is a zero of f of x, then the conjugate a minus bi is also a zero of f of x. So this tells us our conjugate or our complex zeroes imaginary zeroes will always come in pairs. So if we know one, we automatically know a second one as well. So if you look at this example here that I've given, notice here that the degree is 4 again, so we're going to be looking for four zeroes if we include multiplicities there. And I also told you that 2i is a zero of this polynomial. So from that conjugate zeroes theorem, we automatically know that negative 2i is a second zero of this polynomial. So right away we know two different zeroes and so we have to find two more. So feel free to pause the video and look on your graph, see if you can find any zeroes from the graph to complete all of our zeroes. When I look at the graph I do not see any real zeroes so that just means these last two zeroes here are also going to be imaginary. So what we'll have to do is we will have to divide by these factors that we have x minus 2i and x plus 2i in order to find the zeroes, the remaining two zeroes that we have here. Now it's difficult to do synthetic division with imaginary numbers, I actually never do it, I always just do long division. So in order to be able to use all real numbers we are going to have to multiply these two factors together and since they are conjugates all of the i's will cancel out here. So if we multiply there we will get x squared plus 2ix minus 2ix minus 4i squared. So if you look here the 2ix's will cancel out, we're left with x squared minus 4i squared. But if you think back to when we talked about complex and imaginary numbers, i squared is equal to negative 1 and so this becomes x squared plus 4. So rather than dividing separately by x minus 2i and x plus 2i, we will divide by x squared plus 4. So we'll have to use long division here. So let me get a new slide so we can have a little bit of space. Okay with our long division we're going to divide by x squared plus 4 and we're going to divide our original. Now if you remember when we talked about long division we said sometimes to put in columns for the missing degrees, in this case we're missing an x cubed and an x but because this is an even degree and all of these are even there are no odds anywhere, I know that's not going to happen at all. So that's why I didn't do it, you still can if that helps but I am not going to. So to perform this division I think x squared times what will give us x to the fourth, it's another x squared so that will go on top and then I multiply the x squared by each piece here. So x squared times x squared is x to the fourth, x squared times 4 is 4x squared and then we'll subtract and now I repeat x squared from over here times what is 9x squared, well it will be a 9 so I'll put a plus 9 at the top and then multiply that 9 by each of the terms on the left, 9 times x squared is 9x squared and 9 times 4 is 36. So when I subtract everything here cancels out which is what I want because again it was a factor so my remainder is zero. So notice at the top we have our answer and we were trying to find the zeros here and so our zeros will be when x squared plus 9 is equal to zero. Again you can use the quadratic formula or completing the square if that is easier for you but in this case if we just take the square root of each side we get our last two zeros at positive and negative 3i.