Originally Posted by

**joumas** 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.

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Let P(z) = 2z^4 - 4z^3 + 21z^2 - 36z +27

If z[1] = z-3i

then z[2] = z+3i (Conjugate Root Theorem)

Let z[3], z[4] = third and forth zeros respectively

then P(z) = (z - z[1])(z - z[2])(z - z[3])(z - z[4])

= (z - (z-3i))(z - (z+3i))(z - z[3])(z - z[4])

= (3i)(-3i)(z - z[3])(z - z[4])

= (-9i^2)(z - z[3])(z - z[4])

= 9(z - z[3])(z - z[4])

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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[1]*z[2].

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[1] is meant to be z with a subscript 1, etcetera. Not sure what the notation for that is though.