To determine a polynomial given its zeroes, we first note that if the polynomial has real coefficients, then any complex zeros must come in conjugate pairs. Here, the zeroes are given as 2, 3i, and -3i.
1. We first formulate the polynomial by multiplying terms that contain roots. The roots, or zeros, are the values of x for which the polynomial is equal to zero. Hence, we can write our polynomial with roots represented by the equation (X-2)(X-3i)(X+3i).
2. We first work with the complex roots. That is (X-3i)(X+3i). The result is a difference of squares which yields X^2 + 9.
3. We then multiply this result by the term having the real root to get the polynomial. Thus, we get (X-2)(X^2 + 9) which is P(X) = X^3 + 9X - 2X^2 - 18
4. Finally, to get the polynomial in standard form, we rearrange the terms in descending powers of X which gives us P(X) = X^3 - 2X^2 + 9X - 18.
This is the polynomial function with zeros 2, 3i and -3i.