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Question

Sum of common roots of the equations

z3 + 2z2 + 2z + 1 = 0 and z100 + z32 + 1 = 0 is equal to :


A

0

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B

-1

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C

1

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D

None

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Solution

The correct option is B

-1


z3 + 2z2 + 2z + 1 = (z3 + 1) + 2z(z + 1)

= (z + 1){z2 + z + 1} = 0 z = -1, ω, ω2.

Let f(z) = z100 + 1

f(ω) = ω100 + ω64 + 1 = ω + ω2 + 1 = 0

f(ω2) = ω200 + ω64 + 1 = ω2 + ω + 1 = 0

∴ Common roots are ω and ω2 whose sum is -1.


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