<s>[INST] Interval for which $x^{12}-x^9+x^4-x+1>0$ 
The largest interval for which $x^{12}-x^9+x^4-x+1>0$ is :

$$
x^{12}-x^9+x^4-x+1>0\\
x(x^3-1)(x^8+1)+1>0\\
x(x-1)(x^2+x+1)(x^8+1)+1>0
$$
I can see that this is satisfied for $x\in(-\infty,0]\cup[1,\infty)$. But how can I verify the above function is greater than zero when $x\in(0,1)$ ?
 [/INST] $$x^{12}+x^4(1-x^5)+(1-x)>0$$ for $0<x<1$.
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