A cubic with exactly one real root

Let $f(x)=x^3+x-1$. It is continuous, with $f(0)=-1<0$ and $f(1)=1>0$. By the intermediate value theorem, $f$ vanishes at some point of $(0,1)$.

Its derivative $f'(x)=3x^2+1$ is positive everywhere, so $f$ is strictly increasing on $\mathbb{R}$. A strictly increasing function takes each value at most once, so the root is unique.