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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{

Find all zeros of f(x) = 4x^3 + 12x^2 - 33x + 35. Enter the zeros separated by commas. Enter exact values, not decimal approximations.

Learn how to find all zeros of the polynomial function f(x) = 4x^3 + 12x^2 - 33x + 35 using the Rational Root Theorem, synthetic division, and the quadratic formula. A comprehensive solution guide for high school students studying polynomial functions.

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