Let $\alpha, \beta$ be the roots of $x^2 - 2x + 4 = 0$. Find the value of $\alpha^6 + \beta^6$.
Step-by-Step Solution
Key Concept: Use the recurrence $S_n = (\alpha + \beta)S_{n-1} - \alpha\beta \cdot S_{n-2}$ with $S_0 = 2, S_1 = 2$. Build up $S_2, S_3, \dots , S_6$ step by step.
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Correct Answer: 128