Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$.  (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
Question:
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$.  (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
Answer:
Let's think step by step.

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x^3 - 3x^2 + 9x - 9
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$.  (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
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