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Let $\alpha$ and $\beta$ be the roots of $x^2 - 2x - 1 = 0.$ Let $p(x)$ be a cubic polynomial such that \begin{align*} p(\alpha) &= 2 \alpha, \\ p(\beta) &= 2 \beta, \\ p(\alpha + \beta) &= 2 \alpha + 2 \beta, \\ p(\alpha \beta) &= 4. \end{align*}Find $p(x).$

 Feb 20, 2021
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By using polynomial interpolation principles, I found that p(x) = x^3 - 4x^2 + 6x - 1.

 Feb 20, 2021

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