1. **State the problem:** Simplify and analyze the expression $x^3 - x$. 2. **Formula and rules:** To simplify expressions like this, factorization is useful. Recall the difference of cubes and common factor rules. 3. **Factor the expression:** $$x^3 - x = x(x^2 - 1)$$ 4. **Recognize difference of squares:** $$x^2 - 1 = (x - 1)(x + 1)$$ 5. **Complete factorization:** $$x^3 - x = x(x - 1)(x + 1)$$ 6. **Interpretation:** The expression factors into three linear terms, showing roots at $x=0$, $x=1$, and $x=-1$. 7. **Summary:** The factored form is $x(x - 1)(x + 1)$, which is useful for solving equations or analyzing the function's behavior.