1. **State the problem:** Simplify or analyze the polynomial expression $2x^3 - 6x + 1$. 2. **Identify the type of expression:** This is a cubic polynomial with terms $2x^3$, $-6x$, and a constant $1$. 3. **Look for common factors:** Check if there is a greatest common factor (GCF) among the terms. 4. The terms are $2x^3$, $-6x$, and $1$. The GCF of $2$, $-6$, and $1$ is $1$, so no common factor to factor out. 5. **Check for possible factorization:** Try to factor the polynomial if possible. 6. Since it is a cubic polynomial, try factoring by grouping or using the Rational Root Theorem. 7. Test possible rational roots: factors of constant term $1$ over factors of leading coefficient $2$ are $\pm1$, $\pm\frac{1}{2}$. 8. Evaluate $f(1) = 2(1)^3 - 6(1) + 1 = 2 - 6 + 1 = -3 \neq 0$. 9. Evaluate $f(-1) = 2(-1)^3 - 6(-1) + 1 = -2 + 6 + 1 = 5 \neq 0$. 10. Evaluate $f(\frac{1}{2}) = 2(\frac{1}{2})^3 - 6(\frac{1}{2}) + 1 = 2(\frac{1}{8}) - 3 + 1 = \frac{1}{4} - 3 + 1 = -1.75 \neq 0$. 11. Evaluate $f(-\frac{1}{2}) = 2(-\frac{1}{2})^3 - 6(-\frac{1}{2}) + 1 = 2(-\frac{1}{8}) + 3 + 1 = -\frac{1}{4} + 4 = 3.75 \neq 0$. 12. No rational roots found, so the polynomial is not factorable over the rationals. 13. **Final conclusion:** The polynomial $2x^3 - 6x + 1$ is already in simplest form and does not factor nicely over the rationals.