How to write polynomial functions with rational coefficients

Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1 and the zeros 2i and -3+√5

Answer 1

Complex zeroes always come in pairs so another zero is -2i, which, together with 2i, creates the quadratic:

(x+2i)(x-2i)=x²+4.

Similarly, the irrational zero also has a counterpart -3-√5.

The quadratic is (x+3+√5)(x+3-√5)=x²+6x+9-5=x²+6x+4.

The quartic is;

f(x)=(x²+4)(x²+6x+4)=x⁴+6x³+4x²+4x²+24x+16=x⁴+6x³+8x²+24x+16.