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?

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.