00:01
This problem says a polynomial f of x and one of its zeros are given.
00:03
F f of x is a product of linear factors.
00:06
And we see the function we're given here as well as the zero, three plus two i.
00:10
And even though we are told we're only given one of its zeros, we know that this is a complex imaginary form where if we have three plus two i, three minus two i also has to be a zero.
00:20
So we already know some of the linear factors that are going to make up this function, but we need to figure out what the rest are.
00:26
And the way to do that is to look at these two zero.
00:30
In linear factor form, so x minus 3 plus 2i, and then x minus 3 minus 2i.
00:38
And what we can do is divide with these factors after we multiply them together to get rid of the imaginary i values that we see.
00:46
So i'm going to distribute the subtraction for both.
00:49
So we get x minus 3 minus 2i, and then x minus 3 plus 2i.
00:55
And then we're going to multiply these two trinomials together to see what we get as far as a smaller polynomial.
01:00
That maybe we could divide into f of x to start to break it down.
01:04
So x times all three terms will be x squared minus 3x plus 2ix.
01:10
3 times all three terms will be, or negative three times all three terms will be minus 3x plus 9 and then minus 6i.
01:18
And then lastly, negative 2i times everything will give us negative 2i x plus 6i.
01:24
And then negative 2i times positive 2i will be negative 4 i squared so now what we're going to see happen is a lot of our terms cancel out we'll have x squared and then negative 3x minus 3x will give us negative 6x but then we see a positive 2 i x and a negative 2 ix that will cancel themselves out and negative 6 i plus 6 i will also cancel themselves out and then we have plus 9 minus 4 i squared but remember that i squared is negative 1 so that's negative 4 times negative 1 so that's plus 4.
01:55
So our simplified trinomial will be x squared minus 6x plus 13.
02:01
And we know this has to be a factor of our function because it was created from the zeros we were given...