Question

Find a polynomial with real coefficients that has zeros at -2, 4, and 3i

Answer 1

If "a" is a zero of f(x), then (x-a) is a factor of f(x) (CONVERSE OF FACTOR THEOREM)

Answer 2

Moreover, for all polynomials with real coefficients, if a+bi is a zero of it, then a-bi will also be a zero of it, where a and b are real numbers

Answer 3

The root 3i of the equation should be the root of one factor in the form:
(x^2 + 9) = 0 -> x^2 = -9 = 9i^2 --> x = 3i
The polynomial would have the form:
y =(x + 2)(x - 4)(x^2 + 9)
y = (x^2 - 2x - 8)(x^2 + 9)