Multiple choice

If $\alpha$ and $\beta$ are the roots of the equation $x^{2} - x - 1 = 0$, then what is the value of $(\alpha^{4} + \beta^{4})$?

  1. $7$
  2. $0$
  3. $2$

  4. None of the above
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A Correct answer
Explanation

Roots satisfy x^2 = x + 1. Then x^4 = (x+1)^2 = x^2 + 2x + 1 = (x+1) + 2x + 1 = 3x + 2. So alpha^4 + beta^4 = 3(alpha+beta) + 4. Since alpha+beta = 1, sum = 3(1) + 4 = 7.

AI explanation

From the equation, the sum of the roots alpha and beta is 1 and their product is minus 1. Using the identity for the sum of fourth powers, alpha to the fourth plus beta to the fourth equals the square of alpha squared plus beta squared minus 2 times alpha times beta. First, alpha squared plus beta squared equals the square of the sum minus twice the product, which is 1 squared minus 2 times minus 1, giving 3. Substituting this into the fourth power identity gives 3 squared minus 2 times minus 1, which equals 7.

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