Since we are looking for 'real coefficients', complex roots will have to come in pairs.
So with the root of -2i given, we want its conjugate root of 2i.
So the roots are
x = 1
→ x - 1 = 0,
x = - 2i
→ x + 2i = 0, and
x = 2i
→ x - 2i = 0
→ f(x) = (x - 1)(x + 2i)(x - 2i),
which I will expand. Multiply the quantities with the complex roots together first, as terms will cancel, and make the final multiplication easier,
→ f(x) = (x - 1)(x2 - 2xi + 2xi - 4i2). Note that i2 = -1,
→ f(x) = (x - 1)(x2 - 4(-1))
→ f(x) = (x - 1)(x2 + 4). Now multiply these to together,
→ f(x) = x3 + 4x - x2 - 4
→ f(x) = x3 - x2 + 4x - 4
Thank you for posting the question with Wyzant, and have a great night!