Question

How do you find a third degree polynomial given roots `-4` and `4i`?

Answer 1

`x^3+4x^2+16x+64 = 0`

Explanation:

Since the question ask for a third degree polynomial, I am going to assume that you want a polynomial with real coefficients, with `-4i` as the third root, being the complex conjugate of `4i`.

Each zero `a` corresponds to a factor `(x-a)`, so we can write:

`f(x) = (x+4)(x-4i)(x+4i)`

`color(white)(f(x)) = (x+4)(x^2-(4i)^2)`

`color(white)(f(x)) = (x+4)(x^2+16)`

`color(white)(f(x)) = x^3+4x^2+16x+64`

So we can write a cubic equation:

`x^3+4x^2+16x+64 = 0` with the desired roots.