Find a polynomial function​ f(x) of least degree having only real coefficients with zeros of​ 0, 2 i​, and 2+i

Question

Find a polynomial function​ f(x) of least degree having only real coefficients with zeros of​ 0, 2 i​, and 2+i

Answer 1

(x-0)(x-2)(x-i)(x+I)(x-2-i)(x+2+I) imaginary roots come in conjugate pairs, so degree 6 is smallest polynomial possible

x = 2+I, x-2 = I,

square both sides: (x-2)^2 = -1, x^2 -4x +4 +1= 0 = x^2 -4x +5

= (x^2 -2x)(x^2+1)(x^2-(2+I)^2)

= (x^4 -2x^3 +x^2 -2x)(x^2 -4x +5)

= x^6 -2x^5 +13x^4 -16x^3 -3x^2 -10x