*find a polynomial function of the lowest degree with rational coefficients that has the given numbers as some of its zeros -3i,5 *

You have a complex root, x = -3i

Now the thing about complex roots is that they always come in pairs, as complex conjugates.

If one complex root is (a + ib), then the other complex root is (a - ib)

since you have x = -3i, then you must also have x = 3i.

Your lowest polynomial will have three roots, so three factors, (x - 3i), (x + 3i) and (x - 5), giving a cubic function.

The function then is.

(x - 3i)(x + 3i)( x - 5) =

(x^2 - (3i)^2)(x - 5 =

(x^2 + 9)(x - 5) =

x^3 + 9x - 5x^2 - 45 =

**f(x) = x^3 - 5x^2 + 9x - 45**