00:01
We are given this polynomial f of x equals x power 4 plus 55 x squared minus 500 and 76.
00:09
And we are given that negative 8i is a 0 and we have to list the remaining zeros.
00:15
So since negative 8i is a 0 which is a complex root, we see that its conjugate, which is 8i is also a 0 of this polynomial f of x.
00:27
So first we write down these two zeros in factor form.
00:32
That is, we write down as x minus of this negative 8i and also the factor x minus 8i.
00:41
These two are factors of the polymer.
00:43
So this basically equals x minus minus 8i equals x plus 8i and then another factor is x minus 8i.
00:53
So these are basically the factors of this polymer which means the product.
00:58
Of these factors will also be factor of this problem.
01:02
So let's find the product of these two factors.
01:05
That is x plus i times x minus i.
01:09
So we basically multiply these two factors.
01:12
We can do so by using the difference of squares.
01:16
So when we use the difference of squares, which is if we have a plus b times a minus b, and this is equal to a squared minus b squared.
01:27
So this is basically the difference of squares formula.
01:30
We will utilize this to write down in simple form.
01:33
So here a equals x and b equals 8i.
01:39
So therefore when you multiply these two factors, we write down a squared which is x squared minus b squared which is 8i squared.
01:49
And so this equals x squared minus 8i square is 64i squared.
01:57
Remember that in complex number concept, i squared equals negative 1.
02:03
This is i squared equals negative 1.
02:07
So replace i squared by negative 1 into this expression.
02:11
That is 6x squared minus 64 times negative 1...