# Wacky Polynomial

Algebra Level 5

Let $$f(x) = x\left( 4x^2-3 \right) \left( 64x^6-96x^4+36x^2-3 \right)$$. Find the last three digits of the number of distinct positive reals $$x$$ satisfying $$f^{(2014)}(x) = x$$.

Details and Assumptions

• $$f^{(2014)}(x)$$ denotes the function $$f$$ applied $$2014$$ times successively on $$x$$, i.e. $$f^{(2014)}(x) = \underbrace{f ( f ( \cdots ( f(x) )\cdots ))}_{2014 \text{ times}}$$.

• Any non-real / repeating solutions of the equation should be ignored. For example, the roots of the equation (counted with multiplicity) $$x^5-14 x^4 + 86 x^3 - 298 x^2 + 573 x - 468=0$$ are $$\left \{ 3, 3, 4, 2-3i, 2+3i \right \}$$, but its number of distinct real solutions is $$2$$.

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