Find the polinomial with lowest degree having the following zeros : 2 and 2i

The zeroes of a polynomial are the roots of that polynomial. We
also know that the complex solutions come in complex conjugate
pairs.

So, if 2i is a solution of the equation, that means that -2i
is also the solution of the equation.

We also know that a polynomial
could be written as a product of linear factors, if we know it's
solutions.

We'll note the solutions of the polynomial
as:

x1 = 2

x2 =
2i

x3 = -2i

The polynomial
is:

P(x) = a(x - x1)(x - x2)(x -
x3)

We'll substitute x1,x2 and x3:

P(x)
= (x - 2)(x - 2i)(x + 2i)

We'll write the product (x - 2i)(x + 2i)
as a difference of squares:

(x - 2i)(x + 2i) = x^2 +
4

P(x) = (x - 2)(x^2 + 4)

We'll remove
the brackets:

P(x) = x^3 + 4x - 2x^2 -
8

The lowest degree polynomial, having as zeros the
values 2, 2i, -2i, is: P(x) = x^3 - 2x^2 + 4x - 8.