How to factor polynomials of degree 3 or higher efficiently?

For $x^3 - 6x^2 + 11x - 6 = 0$, the rational root theorem suggests testing $\pm 1, \pm 2, \pm 3, \pm 6$. We find $x = 1$ works: $1 - 6 + 11 - 6 = 0$. Then use synthetic division: $$\begin{array}{r|rrrr} 1 & 1 & -6 & 11 & -6 \\ & \downarrow & 1 & -5 & 6 \\ \hline & 1 & -5 & 6 & 0 \end{array}$$ So $(x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3) = 0$, giving $x = 1, 2, 3$. **General strategies for higher-degree polynomials:** 1. Rational root theorem to find first root 2. Synthetic division to reduce degree 3. For quartics, look for quadratic in disguise (e.g., $x^4 - 5x^2 + 6 = 0$) 4. Grouping terms (e.g., $x^3 + 3x^2 + 3x + 1 = (x+1)^3$) 5. Difference/sum of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$