Write the polynomial equations with zero at - 5,0 and 3i

We are given that the zeroes are at

{-5, 0, 3i}.

By the conjugate pair theorem, - 3i, the conjugate of 3i, is automatically a zero for polynomials with real coefficients.

The factors of the polynomial are:

(x+5), x, (x+3i) and (x-3i)

The last two factors multiply to give a real quadratic factor of x^2+9.

Therefore the polynomial is of the form

f (x) = ax (x+5) (x^2+9)

where a is any non-zero real constant.