00:02
We're asked to find a function of lowest degree having real coefficients with zeros of two and six i.
00:09
So to go from zeros to factors, if you think about if two is the zero, then the factor that that comes from is x minus two.
00:20
Think about what a zero is.
00:21
A zero is what makes the factor equal to zero.
00:25
So if i substitute a two back in there, if i put a two right here in place of x, then that factor becomes zero.
00:33
So my factor is x minus two.
00:36
That's for that real zero.
00:38
Now with the six i, imaginary roots or imaginary zeros have to always come in pairs.
00:46
So the first factor is just going to come from the root as it is, positive six i.
00:52
But we know if six i is a factor, then negative six i also has to be a factor because imaginary or non -real solutions come in pairs.
01:06
Let's use non -real.
01:18
So we have to have the conjugate of that too.
01:22
So that means x minus six i is going to be a factor and x plus six i is going to be a factor.
01:30
Now if we can just leave it like that if we want to, but chances are you're going to need to multiply all of this out...