Quintic Quarrel

$y = x^5 + x^4 + 1$

$y = x^5 + x^4 + x^3 - x^3 + 1$

$y = x^3(x^2 + x+ 1) - (x - 1)(x^2 + x + 1)$

$y = (x^2 + x+ 1)(x^3 - x + 1)$

Now, as $(x^2 + x + 1)$ has imaginary roots, $x^3 - x + 1$ is the one with 1 real root

$x^3 - x + 1 = 0$

Now, graph of $x^3 - x$ is

So, graph of $x^3 - x + 1$ will be above graph shifted one unit up. It will intersect x-axis at about -1.32. So,

$\boxed{|Root| = 1.32}$

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Note: I drew graph myself while solving. Above graph from Wolfram Alpha is taken for explaining.

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