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Question

Common roots of the equations z3+2z2+2z+1=0 and z1985+z100+1=0 are

A
ω,ω2
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B
ω,ω3
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C
ω2,ω3
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D
None of these
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Solution

The correct option is A ω,ω2
The given equation z3+2z2+2z+1=0 can be rewritten as (z+1)(z2+z+1)=0. Its roots are 1,ω and ω2.
Let f(z)=z1985+z100+1
Putting z=1. ω and ω2 respectively, we get
f(1)=(1)1985+(1)100+10
Therefore, 1 is not a root of the equation f(z)=0.
Again, f(ω)=ω1985+ω100+1
=(ω3)661ω2+(ω3)33ω+1
=ω2+ω+1=0
Therefore, ω is a root of the equation f(z)=0.
Similarly, f(ω2)=0
Hence, ω and ω2 are the common roots.

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