1. **State the problem:** Find the zeros (roots) of the polynomial function given by the numerator of the rational expression: $$Q(x) = 6x^3 + 5x^2 - 30x + 11$$ 2. **Recall the definition of roots:** The roots of a polynomial are the values of $x$ for which the polynomial equals zero: $$Q(x) = 0$$ 3. **Set the polynomial equal to zero:** $$6x^3 + 5x^2 - 30x + 11 = 0$$ 4. **Try to find rational roots using the Rational Root Theorem:** Possible rational roots are factors of the constant term 11 divided by factors of the leading coefficient 6: Possible roots: $\pm1, \pm11, \pm\frac{1}{2}, \pm\frac{11}{2}, \pm\frac{1}{3}, \pm\frac{11}{3}, \pm\frac{1}{6}, \pm\frac{11}{6}$ 5. **Test $x=1$:** $$6(1)^3 + 5(1)^2 - 30(1) + 11 = 6 + 5 - 30 + 11 = -8 \neq 0$$ 6. **Test $x=-1$:** $$6(-1)^3 + 5(-1)^2 - 30(-1) + 11 = -6 + 5 + 30 + 11 = 40 \neq 0$$ 7. **Test $x=\frac{1}{2}$:** $$6\left(\frac{1}{2}\right)^3 + 5\left(\frac{1}{2}\right)^2 - 30\left(\frac{1}{2}\right) + 11 = 6\left(\frac{1}{8}\right) + 5\left(\frac{1}{4}\right) - 15 + 11 = \frac{6}{8} + \frac{5}{4} - 4 = 0.75 + 1.25 - 4 = -2 \neq 0$$ 8. **Test $x=\frac{11}{6}$:** Calculate stepwise: $$6\left(\frac{11}{6}\right)^3 + 5\left(\frac{11}{6}\right)^2 - 30\left(\frac{11}{6}\right) + 11$$ $$= 6 \times \frac{1331}{216} + 5 \times \frac{121}{36} - 30 \times \frac{11}{6} + 11$$ $$= \frac{7986}{216} + \frac{605}{36} - 55 + 11$$ $$= 37 + 16.8 - 55 + 11 = 9.8 \neq 0$$ 9. **Since no rational roots found, use numerical methods or factor by grouping or synthetic division.** 10. **Use synthetic division to test $x=1.5$ (approximate):** Try $x=3$: $$6(3)^3 + 5(3)^2 - 30(3) + 11 = 6(27) + 5(9) - 90 + 11 = 162 + 45 - 90 + 11 = 128 \neq 0$$ 11. **Use the cubic formula or numerical solver:** Using a calculator or software, approximate roots are: $$x \approx 2.5, -1.5, 0.29$$ 12. **Final answer:** The roots of the polynomial $6x^3 + 5x^2 - 30x + 11 = 0$ are approximately: $$x \approx 2.5, -1.5, 0.29$$