00:01
This problem asks us to determine a polynomial given its zeros, 2 plus i, 2 minus i, 3, and minus 1.
00:08
So that means i can determine its factors.
00:10
So this would be x plus 1, x minus 3, and then i would have x minus 2 plus i and x minus 2 minus i.
00:23
So now i just need to expand all this, multiply it out, and i'll get a polynomial.
00:27
So let's start by multiplying these two factors together.
00:33
So i distribute each of these terms to each one of these terms.
00:37
So what do i get? well, i'll get an x squared.
00:40
I'm going to start with this x, multiplying by everything in the other factor, minus 2x, minus xi, and i'll do the negative 2, minus 2x, plus 4, plus 2i.
00:54
And i'll do the i, so i get plus an xi, minus 2 i and then minus i squared now remember i squared is just equal to negative 1 and because i'm minusing subtracting negative 1 that's the same thing as adding 1 so i'm going to add 1 at the end there so let's combine our like terms now so i have an x squared and then i have minus 2x minus 2x which is going to be minus 4x i've got negative x i plus xi so those will cancel out i've got plus 2 i minus 2i, those will cancel out, and i've got 4 plus 1 gives me plus 5.
01:32
So i've multiplied these two factors together to get this.
01:35
So let's now multiply that by x minus 3 and then by x plus 1...