Find the polynomial of the third degree with integer coefficients and the leading coefficient of one, if the numbers 2 and 1+2i are zeros of the polynomial. Write your answer in simplest form.

Question

Please answer my question as soon as you can!

Answer 1

x=1 + 2i

x-1 = 2i

(x-1)^2 = -4

x^2 -2x +1 = -4

x^2 -2x +5 = 0 if 1+2i is a zero then x^2-2x+5 is a factor. If 2 is a zero, then x-2 is a factor. Multiply them together to get a 3rd degree polynomial.

(x-2)(x^2-2x+5) = x^3 -4x^2 +9x - 10 is a 3rd degree polynomial which has 2 and 1+2i as zeros and leading coefficient = 1, with integer coefficients.

there's also another imaginary zero. they come in conjugate pairs. the 3rd zero is 1-2i.