Final answer:
To completely factor the polynomial p(x), we can divide it by its complex roots to get a quadratic equation, which can be then factored to find the remaining roots. The factors of p(x) are (x - 3i)(x + 3i)(x + 4i)(x - 4i).
Step-by-step explanation:
To completely factor the polynomial p(x) = x⁴ + x³ + 7x² + 9x - 18 into linear functions, we need to find all the zeros of the polynomial. It is given that -3i is a zero, which means that 3i is also a zero since complex roots always come in conjugate pairs.
To find the other two roots, we can use polynomial division or synthetic division. By dividing the polynomial p(x) by x - 3i and x + 3i, we get a quadratic equation: x² + 16 = 0.
Now, we can factor this quadratic equation as (x + 4i)(x - 4i) = 0. Therefore, the factors of p(x) are (x - 3i)(x + 3i)(x + 4i)(x - 4i).