1. **State the problem:** Factor the cubic polynomial $$x^3 - 3x^2 + 2$$. 2. **Recall the factoring approach:** To factor a cubic polynomial, first try to find rational roots using the Rational Root Theorem, then use polynomial division or synthetic division to factor out the root. 3. **Find possible rational roots:** Possible roots are factors of the constant term 2 divided by factors of the leading coefficient 1, so possible roots are $$\pm 1, \pm 2$$. 4. **Test roots:** - For $$x=1$$: $$1^3 - 3(1)^2 + 2 = 1 - 3 + 2 = 0$$, so $$x=1$$ is a root. 5. **Divide the polynomial by $$x-1$$:** Use synthetic division: $$\begin{array}{r|rrrr} 1 & 1 & -3 & 0 & 2 \\ & & 1 & -2 & -2 \\ \hline & 1 & -2 & -2 & 0 \end{array}$$ So the quotient is $$x^2 - 2x - 2$$. 6. **Factor the quadratic $$x^2 - 2x - 2$$:** Use the quadratic formula: $$x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)} = \frac{2 \pm \sqrt{4 + 8}}{2} = \frac{2 \pm \sqrt{12}}{2} = \frac{2 \pm 2\sqrt{3}}{2} = 1 \pm \sqrt{3}$$. 7. **Write the full factorization:** $$x^3 - 3x^2 + 2 = (x - 1)(x - (1 + \sqrt{3}))(x - (1 - \sqrt{3}))$$. **Final answer:** $$\boxed{(x - 1)(x - 1 - \sqrt{3})(x - 1 + \sqrt{3})}$$