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Properties of Conjugate of a Complex Number

The Common ro...

Question

The Common roots of the equations $z^{3} + 2 z^{2} + 2 z + 1 = 0$ , $z^{2014} + z^{2015} + 1 = 0$ are

A

ω, ω^{2}

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B

1, ω, ω^{2}

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C

- 1, ω, ω^{2}

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D

- ω, {- ω}^{2}

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Solution

The correct option is A $ω, ω^{2}$
Clearly $ω, ω^{2}$ are the roots of given equations.

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Similar questions

1, α_{1}, α_{2}, α_{3}

α_{4}

be the roots of

x^{5} - 1 = 0

\frac{ω - α_{1}}{ω^{2} - α_{1}} . \frac{ω - α_{2}}{ω^{2} - α_{2}} . \frac{ω - α_{3}}{ω^{2} - α_{3}} . \frac{ω - α_{4}}{ω^{2} - α_{4}} =

ω

is an imaginary cube root of unity, then a root of
equation

∣ ∣ ∣ ∣ ∣ \begin{matrix} x + 1 & ω & ω^{2} ω & x + ω^{2} & 1 ω^{2} & 1 & x + 2 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0

α_{1}, α_{2}, α_{3}, α_{4}

be the roots of

x^{5} - 1 = 0

\frac{ω - α_{1}}{ω^{2} - α_{1}} \cdot \frac{ω - α_{2}}{ω^{2} - α_{2}} \cdot \frac{ω - α_{3}}{ω^{2} - α_{3}} \cdot \frac{ω - α_{4}}{ω^{2} - α_{4}}

1, ω, ω^{2}

are three cube roots of unity, then

(1 - ω + ω^{2}) (1 + ω - ω^{2})

1, d_{1}, d_{2}, d_{3}, d_{4}

x^{5} = 1

then the value of expression :

E = \frac{ω - d_{1}}{ω^{2} - d_{1}} \cdot \frac{ω - d_{2}}{ω^{2} - d_{2}} \cdot \frac{ω - d_{3}}{ω^{2} - d_{3}} \cdot \frac{ω - d_{4}}{ω^{2} - d_{4}}

ω

is the cube root of unity]

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