Which is greater $x^{3}$ or $x^{2} + x + 2$ for positive values of $x$ ?

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Solution

Consider $x^{3} - (x^{2} + x + 2)$
$= x^{3} - x^{2} - x - 2$
$= x^{3} - 1 - (x^{2} + x + 1)$
We know that $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$
$= (x - 1) (x^{2} + x + 1) - (x^{2} + x + 1)$
$= (x - 2) (x^{2} + x + 1)$
$x^{2} + x + 1 > 0$ for all $x > 0$
the above expression is greater than $0$ for $x > 2$ hence
$x^{3} > x^{2} + x + 2$ for $x > 2$
$x^{3} < x^{2} + x + 2$ for $x < 2$

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