1. **State the problem:** Find all zeros of the cubic polynomial $$f(x) = 4x^3 - 27x^2 - 12x + 35.$$\n\n2. **Recall the zero-finding method:** Zeros of a polynomial satisfy $$f(x) = 0.$$ We will try to find rational roots using the Rational Root Theorem, then factor and solve the remaining quadratic if any.\n\n3. **Apply the Rational Root Theorem:** Possible rational roots are factors of the constant term 35 divided by factors of the leading coefficient 4.\nPossible roots: $$\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4}, \pm 35, \pm \frac{35}{2}, \pm \frac{35}{4}.$$\n\n4. **Test possible roots:** Start with integer candidates.\nEvaluate $$f(1) = 4(1)^3 - 27(1)^2 - 12(1) + 35 = 4 - 27 - 12 + 35 = 0.$$ So, $$x=1$$ is a root.\n\n5. **Divide polynomial by $$x-1$$:** Use synthetic division or polynomial division.\nSynthetic division setup: coefficients are 4, -27, -12, 35.\nBring down 4.\nMultiply 4*1=4, add to -27: -23.\nMultiply -23*1=-23, add to -12: -35.\nMultiply -35*1=-35, add to 35: 0 remainder.\nQuotient polynomial: $$4x^2 - 23x - 35.$$\n\n6. **Solve quadratic $$4x^2 - 23x - 35 = 0$$:** Use quadratic formula $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=4, b=-23, c=-35.$$\nCalculate discriminant: $$\Delta = (-23)^2 - 4(4)(-35) = 529 + 560 = 1089.$$\n\n7. **Calculate roots:** $$x = \frac{23 \pm \sqrt{1089}}{8} = \frac{23 \pm 33}{8}.$$\n\n8. **Find each root:**\n- $$x = \frac{23 - 33}{8} = \frac{-10}{8} = -\frac{5}{4} = -1.25.$$\n- $$x = \frac{23 + 33}{8} = \frac{56}{8} = 7.$$\n\n9. **List all roots in order:** $$-1.25, 1, 7.$$\n\n**Final answer:** The zeros of $$f(x)$$ are $$x = -\frac{5}{4}, 1, 7.$$