If z - 3i is a factor of $\displaystyle P(z) = 2z^4 - 4z^3 + 21z^2 - 36z + 27$, find the remaining factors.

So I did this:

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

Step 2: $\displaystyle P(z) = (z - z_{1})(z - z_{2})(z - z_{3})(z - z_{4})$

Step 3: $\displaystyle P(z) = (z - z_{1})(z - z_{2})(2z^2 + pz +q)$

Step 4: $\displaystyle 2z^4 - 4z^3 + 21z^2 - 36z + 27 = (z - 3i)(z + 3i)(2z^2 + pz +q)$

Step 5: $\displaystyle 2z^4 - 4z^3 + 21z^2 - 36z + 27 = (z^2 + 9)(2z^2 + pz + q)$

Step 6: $\displaystyle 9q = 27$

Step 7: $\displaystyle q = 3$

Step 8: $\displaystyle 9p = -36$

Step 9: $\displaystyle p = -4$

Step 10: Find the factors from the quadratic. $\displaystyle 2z^2 - 4z + 3$

Step 11: $\displaystyle 2(z^2 - 2z + \frac{3}{2})$

Step 12: $\displaystyle 2[(z^2 - 2z + 1) + \frac{3}{2} - \frac{2}{2})]$

Step 13: $\displaystyle 2[(z - 1)^2 + \frac{1}{2})]$

Step 14: $\displaystyle 2[(z - 1)^2 - (\frac{i}{\sqrt{2}})^2)]$

Step 15: $\displaystyle 2[(z - 1 + \frac{i}{\sqrt{2}})(z - 1 - \frac{i}{\sqrt{2}})]$

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.