Multiple choice

lf $\alpha$ and $\beta$ are the roots of the equation $\mathrm{x}^{2}-\mathrm{x}+1 =0$, then $\alpha^{2009}+\beta^{2009}=$

  1. $-1$
  2. $1$
  3. $2$
  4. $-2$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The roots of x^2 - x + 1 = 0 are -omega and -omega^2 (where omega is the cube root of unity). alpha^3 = -1, so alpha^6 = 1. alpha^2009 = (alpha^6)^334 * alpha^5 = alpha^5 = -omega^5 = -omega^2. Similarly, beta^2009 = -omega. The sum is -(omega + omega^2) = -(-1) = 1.

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