```python
def divide_coefficients(dividend: list[int], divisor: list[int]) -> list[list[int]]:
    if not divisor:
        raise ValueError("Divisor cannot be empty")
    if dividend[-1] % divisor[-1]:
        raise ValueError("Leading coefficients must have an exact multiple relationship")

    rest = list(dividend)
    quotient = []
    while len(rest) >= len(divisor):
        factor = rest[-1] // divisor[-1]
        quotient.append(factor)
        for i, term in enumerate(divisor):
            rest[len(rest) - len(divisor) + i] -= factor * term
        while rest and not rest[-1]:
            rest.pop()

    return [quotient, rest[::-1]]
```