1. **State the problem:** Find the roots of the polynomial $$p(x) = x^3 - 4x^2 - 5x + 14$$ using the Rational Root Theorem and synthetic division. 2. **Rational Root Theorem:** Possible rational roots are of the form $$\pm \frac{p}{q}$$ where $p$ divides the constant term 14 and $q$ divides the leading coefficient 1. So possible roots are $$\pm 1, \pm 2, \pm 7, \pm 14$$. 3. **Test possible roots using synthetic division:** - Test $x = -2$: $$\begin{array}{r|rrrr} -2 & 1 & -4 & -5 & 14 \\ & & -2 & 12 & -14 \\ \hline & 1 & -6 & 7 & 0 \\ \end{array}$$ Remainder is 0, so $x = -2$ is a root. 4. **Divide polynomial by $(x + 2)$:** The quotient is $$x^2 - 6x + 7$$. 5. **Find roots of quadratic $x^2 - 6x + 7 = 0$ using quadratic formula:** $$x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1} = \frac{6 \pm \sqrt{36 - 28}}{2} = \frac{6 \pm \sqrt{8}}{2} = \frac{6 \pm 2\sqrt{2}}{2} = 3 \pm \sqrt{2}$$ 6. **Final roots:** $$x = -2, \quad x = 3 + \sqrt{2}, \quad x = 3 - \sqrt{2}$$ These are the actual roots of the polynomial $p(x)$.