To write a degree 3 polynomial $f(x)$ that satisfies the given conditions, follow these steps:

Step 1: Incorporate the given zeros

1. One zero is $-2i$. Since the coefficients must be integers, its conjugate $2i$ must also be a zero.
2. Another zero is $\frac{6}{5}$. To ensure integer coefficients, the corresponding factor is $5x - 6$.

Thus, the zeros are $-2i, 2i,$ and $\frac{6}{5}$.

Step 2: Write the factors

From the zeros, the polynomial can be written as a product of its factors: $f(x) = (x - (-2i))(x - (2i))\left(5x - 6\right).$

Step 3: Simplify the factors

1. Simplify $(x - (-2i))(x - (2i)) = (x + 2i)(x - 2i)$. This is a difference of squares: $(x + 2i)(x - 2i) = x^2 - (2i)^2 = x^2 - (-4) = x^2 + 4.$

2. Combine with the factor $5x - 6$: $f(x) = (x^2 + 4)(5x - 6).$

Step 4: Expand the polynomial

Multiply the factors: $f(x) = (x^2 + 4)(5x - 6).$

Use the distributive property: $f(x) = 5x(x^2 + 4) - 6(x^2 + 4),$ $f(x) = 5x^3 + 20x - 6x^2 - 24.$

Rearrange the terms in standard form: $f(x) = 5x^3 - 6x^2 + 20x - 24.$

Final Answer:

The polynomial is: $f(x) = 5x^3 - 6x^2 + 20x - 24.$

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Would you like a detailed explanation of any step?

Related Questions:

1. What happens if the given polynomial was degree 4 with similar conditions?
2. How do we verify that a polynomial satisfies its zeros?
3. Can the Rational Root Theorem be applied to check integer coefficients?
4. What are the implications of complex conjugate roots for polynomials with real coefficients?
5. How do fractional roots affect polynomial construction?

Tip:

Always ensure all roots (real and complex) are included when creating polynomials to meet the degree requirement.