 Today, I want to look at finding both real and imaginary zeros of a polynomial. So if you look at this first example on your sheet, this is a fourth degree polynomial. So when we notice the degree here of four, based on the fundamental theorem of algebra and the number of zeros theorem, we know that when we are trying to solve this problem, we're going to be looking for four zeros. The number of zeros will match the degree as long as you also look at multiplicities of zeros. So what I suggest doing is first putting this equation into your graph and looking for the zeros from the graph. So pause the video a second and try to find those zeros. For me, when I looked at the graph, I saw a zero at negative seven, and then a second zero at positive four. And both of these zeros went straight through the x axis, giving them each a multiplicity of one. So what that means is we still have two more zeros to find, but these are both going to be imaginary zeros. So the way that we can find those zeros is by reducing our original polynomial using polynomial division. So what I'm going to do on the next page is I'm going to set up a synthetic division problem. So we had zeros at negative four and seven, negative seven and four. And so we're going to reduce and divide by each of those. So remember if you have a zero at negative seven, that gives you a factor at x plus seven. And if you have a zero at four, that gives you a factor at x minus four. So what I'm going to do is divide our original equation by each of those factors. I'm going to use synthetic division, you can do long division if you prefer. But with synthetic division, I'll put the negative seven in our box in the corner. And then I'll put the coefficients in order across. And then I will divide. Now I'm not going to do much explanation here, as I am dividing. So if you need more of an explanation of synthetic division, make sure you take time to actually look at those videos that I've posted. But every time you divide by a factor, the remainder should be zero. So here I just divided by x plus seven. Now I'm going to reduce it even further. I divide by x minus four. So I just leave it right where it is in order to save myself a little bit of space. And once again, I have a remainder of zero. So what I'm left with once I have reduced it is this expression 109. And that is really the quadratic x squared plus zero x plus nine. And I need to find when that's equal to zero. Because if you remember back at the beginning, that's really what I'm trying to find. I'm trying to find the zeros of the function. So as we've talked about, there are multiple ways to solve this, you can use your quadratic formula, you can complete the square. I'm going to just solve it by using square roots. And so if I just take x squared plus nine, set it equal to zero, I can just solve for x by taking the square root of each side and I get x is equal to a positive or negative three I. If I think back to where I was before, I had zeros at negative seven and four. Now I have my last remaining zeros at positive three I and negative three I. That means the factors or the complete factorization of this problem would give me the expression x plus seven x minus four x minus three I x plus three I.