Question

find a polynomial of lowest degree with rational coefficients that has the zeros -3i,4

Answer 1

You know... when a polynomial has a root r, it simply means that (x-r) is a factor. Get it?

Answer 2

Sorta like this... if I want a polynomial with roots 2 and -4, then it simply goes:
(x - 2)(x - -4)
or
(x - 2)(x + 4)

and in the end, just
\[\large x^2 + 2x - 8\]

Answer 3

Remember that complex roots come in pairs. If -3i is a root, then its complex conjugate, +3i, will also be a root.