Multiple choice

$\displaystyle \alpha, : \beta$ are roots of the equation $\displaystyle x^{2} - x + 1 = 0$ then $\displaystyle \alpha^{1027} + \beta^{1027}$ equals

  1. $\displaystyle 1$
  2. $\displaystyle 0$
  3. $\displaystyle -1$
  4. $\displaystyle 1027$
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A 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^1027 + beta^1027 = (-omega)^1027 + (-omega^2)^1027 = -(omega^1027 + omega^2054). Since omega^3 = 1, omega^1027 = omega^1 = omega and omega^2054 = omega^2. Thus, -(omega + omega^2) = -(-1) = 1.

AI explanation

From the given equation, the sum of roots is 1 and the product of roots is 1, so we can write alpha squared equals alpha minus 1. Multiplying by alpha, we find alpha cubed equals alpha squared minus alpha, which equals minus 1. Since the cube of each root is minus 1 and 1027 leaves a remainder of 1 when divided by 3, alpha to the power of 1027 equals alpha cubed to the power of 342 multiplied by alpha, which equals 1 multiplied by alpha, yielding alpha. Applying this to both roots, the expression becomes alpha plus beta, which equals 1.

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