# Minima and polynomials?

Algebra Level 5

Let $$a> 0$$ be a real number. It is known that the following equation in $$x$$ has a real root:

$x^{6}+3ax^{5}+(3a^{2}+3) x^{4}+6ax^{3}+(3a^{2}+3) x^{2}+3ax+1+(ax)^{3}=0.$

Find the minimum possible value of $$1000a$$.

This problem is part of the set ... and polynomials

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