Can someone help me with getting the factors for the following problem? I can't seem to get the other set.
If z-3i is a factor of 2z^4 - 4z^3 + 21z^2 - 36z +27, find the remaining factors.
Let P(z) = 2z^4 - 4z^3 + 21z^2 - 36z +27
If z = z-3i
then z = z+3i (Conjugate Root Theorem)
Let z, z = third and forth zeros respectively
then P(z) = (z - z)(z - z)(z - z)(z - z)
= (z - (z-3i))(z - (z+3i))(z - z)(z - z)
= (3i)(-3i)(z - z)(z - z)
= (-9i^2)(z - z)(z - z)
= 9(z - z)(z - z)
I'm pretty sure this is correct so far but I'm not sure how to get the remaining two factors. I'm not sure if you need to divide P(z) by 9 through long division or divide it by z*z.
If someone would be so kind as to help me get the remaining factors which will likely be conjugates.
Thank you in advance.
Note: z is meant to be z with a subscript 1, etcetera. Not sure what the notation for that is though.