<s>[INST] Prove that $x^4 + x^3 + x^2 + x + 1$ has no roots Prove that $x^4 + x^3 + x^2 + x + 1$ has no roots.
I can see two ways of proving it.
The first one is to notice, that $x^5 - 1 = (x - 1)(x^4 + x^3 + x^2 + x + 1)$. It has the only root 1. And it is not the root of $x^4 + x^3 + x^2 + x + 1$. So, $x^4 + x^3 + x^2 + x + 1$ does not have roots.
Another way is to solve it as a palindromic polynomial. It does not have roots.
But is there any way to directly manipulate the expression to show that it is always greater than zero?
 [/INST] There is a theorem about positive polynomials being expressible as the sum of squares. Here $$f(x)=\left(x^2+\frac x2\right)^2 +\frac 34\left(x+\frac 23\right)^2+\frac 23$$
obtained by successively completing the square. Personally I prefer some of the simpler methods. For example one can observe that the roots of $f(x)$ are the complex fifth roots of $1$, so there are no real zeros.
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