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The problem is asking you to write a degree 3 polynomial \( f(x) \) that has integer coefficients and the given zeros: \( -4i \) and \( \frac{8}{5} \).

### Key Points:
1. Since complex zeros always come in conjugate pairs for polynomials with real coefficients, the conjugate of \( -4i \), which is \( 4i \), must also be a root.
2. The roots of the polynomial are:
   - \( -4i \)
   - \( 4i \)
   - \( \frac{8}{5} \)

### Steps to construct the polynomial:
1. The factors corresponding to these zeros are:
   - \( (x + 4i) \) for \( -4i \)
   - \( (x - 4i) \) for \( 4i \)
   - \( \left(x - \frac{8}{5}\right) \) for \( \frac{8}{5} \)

2. To ensure integer coefficients, multiply \( \left(x - \frac{8}{5}\right) \) by 5:
   \[
   5\left(x - \frac{8}{5}\right) = 5x - 8
   \]

3. Multiply the conjugate complex factors:
   \[
   (x + 4i)(x - 4i) = x^2 + 16
   \]
   (since \( (a + bi)(a - bi) = a^2 + b^2 \)).

4. Now, the polynomial is:
   \[
   f(x) = (x^2 + 16)(5x - 8)
   \]

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

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

Would you like more details or have any questions? Here are some related questions for practice:

1. What is the importance of conjugate pairs in polynomials with real coefficients?
2. How do you expand polynomials with complex numbers?
3. Can you derive a polynomial if only a fraction is given as a zero?
4. How does multiplying by a factor like 5 affect the polynomial?
5. What role does the degree of the polynomial play in determining its structure?

**Tip:** When dealing with fractions as roots, always multiply by the denominator to clear it and ensure integer coefficients.

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 zeros -4i and 8/5. This solution uses the concept of complex conjugates and clearing fractions to ensure integer coefficients, with a step-by-step expansion of the polynomial.

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