1. **State the problem:** Simplify and analyze the expression $x^3 - x$. 2. **Formula and rules:** To simplify, factor the expression by finding common factors. 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. **Explain:** The expression factors into three linear factors, which means the roots (solutions to $x^3 - x = 0$) are $x = 0$, $x = 1$, and $x = -1$. 7. **Summary:** The factored form is $x(x - 1)(x + 1)$, showing the zeros clearly.