1. **State the problem:** Solve the polynomial equation $$x^4 + 6x^3 - 33x^2 - 46x + 72 = 0$$ for all roots. 2. **Use the Rational Root Theorem:** Possible rational roots are factors of 72 divided by factors of 1, i.e., $$\pm1, \pm2, \pm3, \pm4, \pm6, \pm8, \pm9, \pm12, \pm18, \pm24, \pm36, \pm72$$. 3. **Test possible roots:** Substitute values to find roots. - Test $x=2$: $$2^4 + 6(2)^3 - 33(2)^2 - 46(2) + 72 = 16 + 48 - 132 - 92 + 72 = -88 \neq 0$$ - Test $x=3$: $$3^4 + 6(3)^3 - 33(3)^2 - 46(3) + 72 = 81 + 162 - 297 - 138 + 72 = -120 \neq 0$$ - Test $x=1$: $$1 + 6 - 33 - 46 + 72 = 0$$ so $x=1$ is a root. 4. **Divide polynomial by $(x-1)$:** Use synthetic division: Coefficients: 1, 6, -33, -46, 72 Bring down 1. Multiply 1 by 1, add to 6: 7. Multiply 7 by 1, add to -33: -26. Multiply -26 by 1, add to -46: -72. Multiply -72 by 1, add to 72: 0. Quotient polynomial: $$x^3 + 7x^2 - 26x - 72$$ 5. **Factor cubic:** Try rational roots again. Test $x=2$: $$8 + 28 - 52 - 72 = -88 \neq 0$$ Test $x=3$: $$27 + 63 - 78 - 72 = -60 \neq 0$$ Test $x=4$: $$64 + 112 - 104 - 72 = 0$$ so $x=4$ is a root. 6. **Divide cubic by $(x-4)$:** Coefficients: 1, 7, -26, -72 Bring down 1. Multiply 1 by 4, add to 7: 11. Multiply 11 by 4, add to -26: 18. Multiply 18 by 4, add to -72: 0. Quotient polynomial: $$x^2 + 11x + 18$$ 7. **Factor quadratic:** $$x^2 + 11x + 18 = (x + 9)(x + 2)$$ 8. **Roots:** $$x = 1, 4, -9, -2$$ 9. **Order roots from smallest to largest:** $$x = -9, -2, 1, 4$$ **Final answer:** $$\boxed{-9, -2, 1, 4}$$