 In this video, we'll look at finding the complex zeros of a polynomial. We'll consider as our example f of x equals x to the fourth plus 7x cubed minus 3x squared minus 113x minus 204. Recall that we have a relationship between the number of complex zeros and the degree of the polynomial. And so in this case, we know we have four complex zeros because we're dealing with a fourth degree polynomial. We also have a relationship between the zeros and the x-intercepts on our graph. And so let's start by graphing this function. When we graph this, we see we have zeros at negative 3 and at 4 because we have x-intercepts at negative 3, 0 and 4, 0. We know we should have two more zeros. In this case, those zeros are going to be imaginary. To find those zeros, we're going to start by factoring out our two known factors of x plus 3 and x minus 4 out of our quartic polynomial. We'll begin with the negative 3. And so we're going to factor out x plus 3 out of our quartic polynomial. To do that, we're going to use synthetic division. And so recall, we're going to list our 0, negative 3, and the coefficients of our polynomial, which were 1, 7, negative 3, negative 113, and negative 204. To do the synthetic division, we're going to begin by dropping down the negative, or excuse me, dropping down the 1. We multiply the 1 by negative 3 and add the result to the 7. And so now we have 4. We're going to multiply the 4 by negative 3 to get negative 12. And we add that to the negative 3 to get negative 15. We multiply the negative 15 by negative 3 to get 45. And we add the 45 to the negative 113 to get negative 68. We're going to multiply the negative 68 by negative 3 to get 204, and we add that to the negative 204, and we have our remainder of zero as expected. We now can write our quartic polynomial as a product of a linear factor and a cubic factor. Our linear factor is going to be the x plus 3. And to find the coefficients of our cubic factor, we look at the result of our synthetic division. We're going to use the 1, the 4, the negative 15, and the negative 68, and we have x cubed plus 4x squared minus 15x minus 68. We want a factor in the cubic polynomial further now, and so we'll use the zero, or 4, and go through the synthetic division process again. So we're going to take the coefficients again, 1, 4, negative 15, and negative 68, and our zero, or 4. And as we go through the synthetic division, we drop down the 1. We multiply the 1 by 4 to get 4, add that to the 4 that was already there to get 8. We multiply the 4 by 8 to get 32, adding that to the negative 15 gives us 17. We multiply the 4 by 17 to get 68, and we add the negative 68 and 68 to get our remainder of zero. Our polynomial then will factor further as x plus 3 times, and we're breaking down the cubic polynomial as x minus 4 times the quadratic polynomial, x squared plus 8x plus 17. Again, our 1, 8, and 17 coefficients are coming from the result of our synthetic division. We now know that our complex zeros are going to come from our quadratic polynomial, and so we'll set x squared plus 8x plus 17 equal to zero. We can solve this using quadratic formula. We have negative 8 plus or minus the square root of 8 squared minus 4 times 1 times 17 over 2 times 1. We can reduce this, the 8 squared is 64, negative 4 times 1 times 17 is negative 68, the 64 minus 68 will be reduced to negative 4, and the square root of negative 4 is 2i. We can then break down the fraction. We have negative 8 divided by 2 is 4, and 2i divided by 2 is i, and so our four complex zeros are negative 3, 4, negative 4 plus i, and negative 4 minus i.