Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$

Question

Does the equation $x^{3} - x - p = 0$ always has a solution in $\mathbb{R}$ for all $p \in \mathbb{R}$.

Is there a real solution for $x$ for each real number $p$?

I am new to the theory of cubic equations.

Answer 1

Yes. The function $x^3 - x - p$ is continuous, it's negative for large negative values of $x$ and positive for large positive values of $x$, so it must be zero somewhere in between, by the intermediate value theorem.