00:01
So in this problem, we're being asked to find a polynomial f of x with real coefficients that has, that is degree 4, and has the following zeros, negative 4 plus 4i, that has a multiplicity of 2.
00:12
Well, remember, because it's a fourth degree polynomial, we should have 4 zeros.
00:16
Now, the multiplicity of 2 means when we write this as a factor, it'll be raised to the second power.
00:22
Well, notice that our 0 is a complex number.
00:25
So if negative 4 plus 4i is one of our zeros, its complex conjugate will be raised.
00:30
As well.
00:31
So it'll be negative 4 minus 4i.
00:33
And it will have the same multiplicity.
00:35
So it will have a multiplicity of 2 as well.
00:38
Okay.
00:39
Now that we know our zeros, we can go to our factored form.
00:42
So to go to our factored form, remember, we take the opposite of the zero.
00:46
So in other words, for our first one, we'll have max minus negative 4 plus 4i.
00:52
And for our second zero, we'll have x minus the quantity of negative 4 minus 4i.
00:58
Now here's when the multiplicity comes in.
01:00
Because it has a multiplicity of two, each of these factors will be raised to the second power.
01:05
So now we just need to simplify.
01:07
Well, first off, let's get rid of those parentheses inside our factors.
01:11
So in other words, we're going to have the quantity of x plus four minus four i and then, oh, being squared, and then we'll have the quantity of x minus, or sorry, excuse me, plus four, plus four i.
01:24
Again, it's being square.
01:26
Well, remember, to square it means to multiply it by itself.
01:29
So i'm going to leave a some room here.
01:30
So we would really have x plus 4 minus 4 i getting multiplied by x plus 4 minus 4 i.
01:38
And for the second one we'll have x plus 4 plus 4 i as well as x plus 4 plus 4 i.
01:45
Now i'm going to give us a little hint.
01:48
Typically when we've just multiplied the ones with the conjugates, it's going to end up being with nice terms.
01:53
So i'm going to group those ones together.
01:54
So in other words, i'm going to group x plus 4 minus 4i with the first x plus 4.
02:00
Plus 4i, and then i'm going to group the second ones together, because when we multiply in both together, we're going to get the same result.
02:07
So this way, we only have to do it once.
02:12
All right, so now let's go ahead and multiply our first two terms here.
02:16
So to do this, we're going to use the distributed property.
02:19
So x times x is x squared.
02:22
X times four is positive 4x.
02:25
X times 4i is positive 4i x.
02:28
Now we'll move to the next term.
02:30
4 times x is positive 4x...