Interval for which $x^{12}-x^9+x^4-x+1>0$

Question

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 $x\in(0,1)$ ?

Answer 1

$$x^{12}+x^4(1-x^5)+(1-x)>0$$ for $0<x<1$.