What is the simplest polynomial equation that can be written that has roots 4 and 1+2i

p (x) = x^3 - 6x^2 + 13x - 20

Step-by-step explanation:

We presume you want a polynomial with real coefficients. If so, then there is another complex root, the conjugate of the one given. With three roots, the polynomial will be 3rd degree:

p (x) = (x - 4) (x - 1+2i) (x - 1-2i) = (x - 4) ((x-1) ^2 - (2i) ^2) = (x - 4) (x^2 - 2x + 5)

p (x) = x^3 - 6x^2 + 13x - 20