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Algebra Difficulty 4.8 AIME Find the answer

Example 3. When x=1+19942x=\frac{1+\sqrt{1994}}{2}, the value of the polynomial
(4x3−1997x−1994)2001 \left(4 x^{3}-1997 x-1994\right)^{2001}

is:

Pick one

Solution

Analyzing direct substitution, it is certainly not the best strategy. From the given, we have 2x−1=19942 x-1=\sqrt{19} 94. Squaring and rearranging gives
4x2−4x−1993=0.  4 x^{2}-4 x-1993=0 \text {. }

And
4x3−1997x−1994=(x−1)(4x2−4x−1993)−1=−1. \begin{array}{l} 4 x^{3}-1997 x-1994 \\ =(x-1)\left(4 x^{2}-4 x-1993\right)-1 \\ =-1 . \end{array}

Therefore, the desired result is (−1)2001=−1(-1)^{2001}=-1.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.