The number of common roots of the equations $x^{3} + 2 x^{2} + 2 x + 1 = 0$ and $x^{2012} + x^{2014} + 1 = 0$ is

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answer is B.

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Detailed Solution

$\begin{array}{l} x^{3} + 2 x^{2} + 2 x + 1 = 0 \\ - 1 \begin{array}{l} 1 2 2 1 \\ 0 - 1 - 1 - 1 \end{array} \\ 1 1 1 0 \\ (x + 1) (x^{2} + x + 1) = 0 \\ α = - 1, β = ω, δ = ω^{2} a r e r o o t s o f t h e e q u a t i o n x^{3} + 2 x^{2} + 2 x + 1 = 0 \\ x^{2012} + x^{2014} + 1 = 0 \\ β = ω \Rightarrow x^{2012} + x^{2014} + 1 = ω^{2012} + ω^{2014} + 1 = ω^{2010} . ω^{2} + ω^{2013} . ω + 1 \\ 1. ω^{2} + 1. ω + 1 = 0 \\ s i m i l a r l y δ = ω^{2} s a t i s f y t h e e q u a t i o n x^{2012} + x^{2014} + 1 = 0 \\ n o . o f c o m m o n r o o t s = 2 (ω, ω^{2}) \end{array}$