Multiple choice

lf $\alpha,\ \beta$ are the roots of the equation $\mathrm{x}^{2}+\mathrm{x}+1 =0$ then $\alpha^{4}+\beta^{4}=$

  1. $ -1$
  2. $ 0$
  3. $1$
  4. $2$
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A Correct answer
Explanation

The roots of x^2 + x + 1 = 0 are omega and omega^2, where omega^3 = 1. Thus alpha^4 + beta^4 = omega^4 + (omega^2)^4 = omega + omega^8 = omega + omega^2. Since 1 + omega + omega^2 = 0, omega + omega^2 = -1.

AI explanation

From the equation x^2 + x + 1 = 0, dividing by x gives x + 1 + 1/x = 0, so x + 1/x = -1. Squaring both sides results in x^2 + 2 + 1/x^2 = 1, meaning x^2 + 1/x^2 = -1. Squaring this again gives x^4 + 2 + 1/x^4 = 1, so x^4 + 1/x^4 = -1. Since alpha and beta are the roots of the original equation, substituting them for x gives alpha^4 + beta^4 = -1.

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