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Algebra Difficulty 4.8 AIME Find the answer

Let p(x)=x2−x+1p(x)=x^{2}-x+1. Let α\alpha be a root of p(p(p(p(x))))p(p(p(p(x)))). Find the value of (p(α)−1)p(α)p(p(α))p(p(p(α)))(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha)))

A number or a short expression. Spacing and $ signs are ignored.

Solution

Since (x−1)x=p(x)−1(x-1) x=p(x)-1, we can set (p(α)−1)p(α)p(p(α))p(p(p(α)))=(p(p(α))−1)p(p(α))p(p(p(α)))=(p(p(p(α)))−1)p(p(p(α)))=p(p(p(p(α))))−1=−1(p(\alpha)-1) p(\alpha) p(p(\alpha)) p(p(p(\alpha))) =(p(p(\alpha))-1) p(p(\alpha)) p(p(p(\alpha))) =(p(p(p(\alpha)))-1) p(p(p(\alpha))) =p(p(p(p(\alpha))))-1 =-1

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