00:01
What we're being asked to do is to find the polynomial function that is fourth degree and has the zeros 3i and 2 minus 5i.
00:07
Well, remember, if the polynomial is fourth degree, that means it needs to have four zeros.
00:12
Notice that the two zeros that you're given right now are complex numbers.
00:16
And remember, anytime you have a complex number as a zero, its complex conjugate is a zero as well.
00:21
Therefore, because 3i is a 0, we know that negative 3i will also be a 0.
00:25
And because 2 minus 5i will be a 0 is complex conjugate 2 plus 5.
00:30
I will also be a zero.
00:32
Perfect.
00:33
So now that we have the zeros, we can go ahead and find the factors for this polynomial.
00:37
Because remember, to go from the zeros to the factors, you use the opposite sign.
00:40
So, because 3i is a zero, x minus 3i is a factor.
00:44
Then i'm going to skip to negative 3i, which would mean x plus 3i is a factor.
00:49
So notice you switch the signs.
00:50
Now, because 2 minus 5i is a 0, that means x minus 2 plus 5i will be a factor.
00:56
Notice you switch both sides.
00:58
And because 2 plus 5i is a 0, that means x minus 2 minus 5i will be a factor.
01:05
Now, because they want this polynomial to have all real coefficients, what that means is we need to multiply these terms out.
01:11
So i'm going to multiply the terms with the complex conjugates together.
01:15
So we'll start by multiplying x minus 3i by x plus 3i.
01:18
Well, notice that's one of our special products.
01:21
It's the product of the sum and difference of two terms.
01:23
So the result will just be two terms.
01:25
The first term is the square of the first term, which is x squared, and then we'll subtract this by the square of our second term, which would be 9i squared.
01:34
Then when we multiply x minus 2 plus 5i by x minus 5i, we're actually just going to have to multiply this up.
01:41
So we're going to have x times x, which is x squared, x times negative 2, that's negative 2x, and x times negative 5i is negative 5 ix.
01:50
Then for the negative 2, negative 2 times x is negative 2x, negative 2 times negative 2 is positive 4, and negative 2, 2 2 ,000 ,000 ,000 ,000, and negative 2, times negative 5i is positive 10i...