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The problem asks for a degree 3 polynomial \( f(x) \) with integer coefficients, given that the zeros are \( -4i \) and \( \frac{8}{5} \).

### Step-by-Step Solution:

1. **Identify all the roots**:
   - Since \( -4i \) is a root, its complex conjugate \( 4i \) must also be a root (because the polynomial has integer coefficients).
   - The third root is \( \frac{8}{5} \).

2. **Write the factors corresponding to the roots**:
   - For the roots \( -4i \) and \( 4i \), their corresponding factor is \( (x - (-4i))(x - 4i) = (x + 4i)(x - 4i) = x^2 + 16 \) (since \( (a - b)(a + b) = a^2 - b^2 \)).
   - For the root \( \frac{8}{5} \), the corresponding factor is \( \left(x - \frac{8}{5}\right) \), but to make the coefficients integers, we multiply by 5 to get \( (5x - 8) \).

3. **Form the polynomial**:
   The polynomial is the product of the factors:
   \[
   f(x) = (x^2 + 16)(5x - 8)
   \]

4. **Expand the polynomial**:
   Expanding \( (x^2 + 16)(5x - 8) \):
   \[
   f(x) = x^2(5x - 8) + 16(5x - 8)
   \]
   \[
   = 5x^3 - 8x^2 + 80x - 128
   \]

Thus, the polynomial is:
\[
f(x) = 5x^3 - 8x^2 + 80x - 128
\]

Would you like further details or clarification on any of the steps?

### Related Questions:
1. What happens if the polynomial has non-integer coefficients?
2. How do we handle cases where there are no complex roots?
3. Can you explain why complex conjugates must appear in polynomials with real coefficients?
4. How does changing the degree of a polynomial affect its structure?
5. What are the general rules for constructing a polynomial from given roots?

### Tip:
Always remember that if a polynomial has real (or integer) coefficients and a complex root, its complex conjugate must also be a root.

Write a polynomial f(x) that satisfies the given conditions. Degree 3 polynomial with integer coefficients with zeros -4i and 8/5.

Learn how to construct a degree 3 polynomial with integer coefficients given the roots -4i, 4i, and 8/5. This step-by-step solution uses polynomial multiplication and the conjugate root theorem, suitable for Grades 10-12.

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