Prove that $x^3 + x^2 = 1$ has no rational solutions?

By the rational root theorem, a rational root would have to be $x=1$ or $x=-1$, but neither works.

Let's assume $x = p/q$. $p$ and $q$ integers without a common factor. Then, $$ p^{3} + p^{2}q = q^{3} $$

It's is only satisfied whenever $p$ and $q$ are simultaneously even. It contradicts the initial hypothesis that we can set $x = p/q$ where $p$ and $q$ has not common factors. $$ \mbox{Then,}\quad x \not\in {\mathbb Q} $$