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Complex Roots of Unity

XAT 2019 · Quantitative Ability & Data Interpretation · Medium · Algebra

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If x2+x+1=0x^2 + x + 1 = 0, then x2018+x2019x^{2018} + x^{2019} equals which of the following:

  1. A.

    x+1x + 1

  2. B.

    xx

  3. C.

    −x-x

  4. D.

    x−1x - 1

  5. E.

    1

Answer

C

Explanation

Since x2+x+1=0x^2 + x + 1 = 0, xx is a cube root of unity ω\omega. Therefore x3=1x^3 = 1. x2018=x3×672+2=x2x^{2018} = x^{3 \times 672 + 2} = x^2. x2019=x3×673=1x^{2019} = x^{3 \times 673} = 1. x2018+x2019=x2+1x^{2018} + x^{2019} = x^2 + 1. From x2+x+1=0⇒x2+1=−xx^2 + x + 1 = 0 \Rightarrow x^2 + 1 = -x.

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