Multiple choice

If α and β are the roots of the equation x2 + x + 1 = 0, then the value of α3 + β3 is

  1. -1

  2. 0

  3. 1

  4. 2
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D Correct answer
Explanation

The roots of x^2 + x + 1 = 0 are the complex cube roots of unity, omega and omega^2. Since omega^3 = 1, alpha^3 + beta^3 = omega^3 + (omega^2)^3 = 1 + 1 = 2.

AI explanation

Using Vieta's formulas for the equation x^2 + x + 1 = 0, the sum of the roots (alpha + beta) is -1 and the product of the roots (alpha * beta) is 1. Apply the algebraic identity for the sum of cubes, which states alpha^3 + beta^3 = (alpha + beta)^3 - 3 * alpha * beta * (alpha + beta). Substituting the known values gives (-1)^3 - 3 * (1) * (-1), resulting in -1 + 3. The final value is 2.

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