1. **State the problem:** Find the complex zeros of the polynomial function $$f(x) = x^3 - 12x^2 + 49x - 58$$ and write $$f(x)$$ in factored form. 2. **Use the Rational Root Theorem** to test possible rational roots, which are factors of 58 (constant term) over factors of 1 (leading coefficient): $$\pm1, \pm2, \pm29, \pm58$$. 3. **Test $$x=1$$:** $$f(1) = 1 - 12 + 49 - 58 = -20 \neq 0$$ 4. **Test $$x=2$$:** $$f(2) = 8 - 48 + 98 - 58 = 0$$ So, $$x=2$$ is a root. 5. **Divide $$f(x)$$ by $$x-2$$ using synthetic division:** $$\begin{array}{r|rrrr} 2 & 1 & -12 & 49 & -58 \\ & & 2 & -20 & 58 \\ \hline & 1 & -10 & 29 & 0 \\ \end{array}$$ The quotient is $$x^2 - 10x + 29$$. 6. **Find zeros of the quadratic $$x^2 - 10x + 29$$ using the quadratic formula:** $$x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 29}}{2 \cdot 1} = \frac{10 \pm \sqrt{100 - 116}}{2} = \frac{10 \pm \sqrt{-16}}{2}$$ 7. Simplify the square root of negative number: $$\sqrt{-16} = 4i$$ 8. So the complex roots are: $$x = \frac{10 \pm 4i}{2} = 5 \pm 2i$$ 9. **List all complex zeros:** $$\boxed{2, 5 + 2i, 5 - 2i}$$ 10. **Write $$f(x)$$ in factored form using the roots:** $$f(x) = (x - 2)(x - (5 + 2i))(x - (5 - 2i))$$ 11. **Multiply the complex conjugate factors:** $$ (x - (5 + 2i))(x - (5 - 2i)) = \left(x - 5 - 2i\right)\left(x - 5 + 2i\right) = (x - 5)^2 - (2i)^2 = (x - 5)^2 + 4 $$ 12. Expand: $$ (x - 5)^2 + 4 = x^2 - 10x + 25 + 4 = x^2 - 10x + 29 $$ 13. **Final factored form:** $$\boxed{f(x) = (x - 2)(x^2 - 10x + 29)}$$