<s>[INST] Finding domain of $\sqrt{x^{12} - x^9 + x^4 - x + 1}$ How to find the domain of this function $\sqrt{x^{12} - x^9 + x^4 - x + 1}$?
Its really hard to find all 12 roots and plot them or use tricks like "wavy curve" to check where the expression in the square root is positive . But I thought (i'm not sure) if we can prove that the function is positive for all value then the job would be done, although i haven't been able to do it. It would be helpful if someone can solve this .
 [/INST] $$x^{12}-x^9+x^4-x+1=\frac{1}{2}\bigg[2x^{12}-2x^9+2x^4-2x+2\bigg]$$
$$=\frac{1}{2}\bigg[(x^{12}-2x^9+x^6)+\bigg(x^{12}-x^6+\frac{1}{4}\bigg)+x^4+\bigg(x^4-x^2+\frac{1}{4}\bigg)+(x^2-2x+1)+\frac{1}{2}\bigg]$$
$$=\frac{1}{2}\bigg[(x^6-x^3)^2+\bigg(x^6-\frac{1}{2}\bigg)^2+x^4+\bigg(x^2-\frac{1}{2}\bigg)^2+(x-1)^2+\frac{1}{2}\bigg]>0\forall x\in\mathbb{R}.$$
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