If z - 3i is a factor of , find the remaining factors.

So I did this:

Step 1: if z - 3i is a factor, then z + 3i is a factor.

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

Step 7:

Step 8:

Step 9:

Step 10: Find the factors from the quadratic.

Step 11:

Step 12:

Step 13:

Step 14:

Step 15:

So this is the answer I got, but the book says just that these are the two remaining factors without the leading 2, as follows:

(z - 1 + \frac{i}{\sqrt{2}})(z - 1 - \frac{i}{\sqrt{2}}

So did I get it wrong or is the 2 not really classified as part of the factor for some reason? I can't understand why the 2 isn't included in the answer in the back of the book.

David.