Find the cubic polynomial, in $x,$ with integer coefficients that has $\cos 20^\circ$ as a root.  The coefficient of $x^3$ should be positive, and the coefficients should have no common factor other than 1.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the cubic polynomial, in $x,$ with integer coefficients that has $\cos 20^\circ$ as a root.  The coefficient of $x^3$ should be positive, and the coefficients should have no common factor other than 1.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
8x^3 - 6x - 1
Find the cubic polynomial, in $x,$ with integer coefficients that has $\cos 20^\circ$ as a root.  The coefficient of $x^3$ should be positive, and the coefficients should have no common factor other than 1.
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