00:01
In this question, we are asked to write a fourth degree polynomial with real coefficients with some given roots.
00:09
So we want the following roots, positive 3, positive 2, and i, which is just the square root of negative 1.
00:21
And what we can keep in mind is that any polynomial with real coefficients that has i as a root will also have negative i as a root.
00:29
So here we can take these four roots and cook up a fourth degree polynomial as follows.
00:37
So i want to do, i want to write the factors corresponding to each root.
00:44
So positive 3 is going to correspond to a factor of x minus 3.
00:48
Remember the signs always switch.
00:51
Positive 2 will correspond to a factor of x minus 2.
00:58
I will correspond to a factor of x minus i.
01:06
And negative i is going to correspond to a factor of x minus negative i which is just x plus i and now we can multiply these factors out so first let's multiply out these two factors so if i foil this i get x squared minus 2x minus 3x and then negative 3 times negative 2 gives me plus 6 which is equal to x squared minus 5x plus six and i'm going to bring these roots down.
01:50
All right.
01:56
Now maybe i'll multiply these final two roots.
02:03
So i'm going to carry the x squared minus 5x plus 6 along.
02:07
Now when i foil this out, x squared, i get plus i x and then i also get minus i x and then i times i is negative 1.
02:22
So negative times positive i is positive one...