Given 3 i is a root of 2 z4 3 z3 39z2 120 z 50 0 find

Given 3 + i is a root of 2 z^4 - 3 z^3 - 39z^2 + 120 z - 50 = 0, find the other roots with all workings shown.

Solution

If 3 + i is a root , then 3-i is also a root .

And (z-3-i)(z-3+i) = z²-6z + 10

Now we divide the given polynomial by z²-6z +10 to reduce it .

And on dividing we get 2z²+9z -5 = (2z-1)(z+5)

2z-1=0   , z+5=0

z=1/2 , -5

Therefore the roots are z=3+i , 3-i ,-5,1/2