To find all zeros of the polynomial function $f(x) = 4x^3 + 12x^2 - 33x + 35$, we can use several techniques, such as the Rational Root Theorem, synthetic division, or factoring if possible. Let's go through the process step-by-step:

1. Identify potential rational roots using the Rational Root Theorem:

   • The Rational Root Theorem suggests that possible rational roots are the factors of the constant term (35) divided by the factors of the leading coefficient (4).
   • Factors of 35 are $\pm 1, \pm 5, \pm 7, \pm 35$.
   • Factors of 4 are $\pm 1, \pm 2, \pm 4$.
   • Possible rational roots are therefore: $\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 5, \pm \frac{5}{2}, \pm \frac{5}{4}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4}, \pm 35, \pm \frac{35}{2}, \pm \frac{35}{4}$

2. Test these roots using synthetic division to determine which are actual roots.

After testing possible values, the root $x = \frac{5}{2}$ works, as substituting $x = \frac{5}{2}$ into the polynomial gives zero.

3. Divide the polynomial by $x - \frac{5}{2}$ (or equivalently multiply by $2x - 5$) to factor it.

   After division, we find: $f(x) = (2x - 5)(2x^2 + 10x - 7)$

4. Solve the quadratic $2x^2 + 10x - 7 = 0$ using the quadratic formula: $x = \frac{-10 \pm \sqrt{(10)^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2}$ $x = \frac{-10 \pm \sqrt{100 + 56}}{4}$ $x = \frac{-10 \pm \sqrt{156}}{4}$ $x = \frac{-10 \pm 2\sqrt{39}}{4}$ [ x = \frac{-5 \pm \sqrt{