Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 2, and has the zeros −3, 0, (2 + √3), and (1 + 4i).

Answer 1

x = 2+sqr3

x-2 = sqr3

(x-2)^2 = 3

x^2 -4x +4 = 3

x^2 -4x +1 is one factor

x = 1+4i

x-1 = 4i

(x-1)^2 =-16

x^2 -2x +1 +16 = 0

x^2 -2x +17 is another factor

two more factors are

x+3 and x

multiply the 4 factors together to get a 6th degree polynomial function

f(x) = (x^2 +3x)(x^2-2x+17)(x^2-4x +1) which is the least degree polynomial with rational coefficients