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To find the remaining zeros of the function \( h(x) = 4x^4 + 5x^3 + 30x^2 + 45x - 54 \), given that one zero is \( -3i \), we can proceed as follows:

### Step 1: Recognize Conjugate Pairs
Since \( -3i \) is a zero and the polynomial has real coefficients, its complex conjugate \( 3i \) must also be a zero.

### Step 2: Form a Quadratic Factor
If \( -3i \) and \( 3i \) are zeros, then \( x^2 + 9 \) is a factor of \( h(x) \) (since the product of \((x + 3i)(x - 3i)\) yields \( x^2 + 9 \)).

### Step 3: Divide the Polynomial by \( x^2 + 9 \)
To find the other factors, divide \( h(x) \) by \( x^2 + 9 \). Let's perform this polynomial division.

### Step 4: Polynomial Division
Dividing \( h(x) = 4x^4 + 5x^3 + 30x^2 + 45x - 54 \) by \( x^2 + 9 \) will yield a quotient polynomial, which will be a quadratic. Solving this quotient will give us the remaining real zeros.

Upon performing this division:

\[
h(x) = (x^2 + 9)(4x^2 + 5x - 6)
\]

### Step 5: Factor the Quadratic Polynomial
Now, factor \( 4x^2 + 5x - 6 \):
\[
4x^2 + 5x - 6 = (4x - 3)(x + 2)
\]

### Step 6: List All Zeros
The zeros of \( h(x) \) are:
1. \( -3i \) (given),
2. \( 3i \) (conjugate),
3. \( \frac{3}{4} \) (from \( 4x - 3 = 0 \)),
4. \( -2 \) (from \( x + 2 = 0 \)).

### Final Answer
The remaining zeros of \( h(x) \) are:
\[
3i, \frac{3}{4}, -2
\]

Would you like further explanation on any steps?

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Here are 5 related questions for deeper understanding:
1. How do we confirm that complex roots come in conjugate pairs for polynomials with real coefficients?
2. What methods can be used to factor a polynomial of higher degrees systematically?
3. How does synthetic division work compared to polynomial division for finding zeros?
4. How can we verify that all roots have been found in a polynomial equation?
5. What are the applications of complex roots in real-world scenarios?

**Tip:** When a polynomial has complex roots, they often appear in conjugate pairs if the polynomial has real coefficients.

Use the given zero to find the remaining zeros of the function. h(x) = 4x^4 + 5x^3 + 30x^2 + 45x - 54; zero: -3i

Discover how to find the remaining zeros of the polynomial function h(x) = 4x^4 + 5x^3 + 30x^2 + 45x - 54 given a complex zero -3i. This problem involves polynomial division and the concept of complex conjugate pairs.

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