Find a polynomial function with real coefficients that has the given zeros. (There are many correct answers.) 4, 4, 1 + i

Answer 1

x=1+i

x also = 1-i, as imaginary zeros come in conjugate pairs

x-1= i

x^2-2x +1 = -1

x^2-2x +2 = 0

x^2-2x +2 is a factor

(x-4)(x-4) = (x-4)^2 is the other factor

f(x) = (x-4)^2(x^2-2x+2)

(x^2-8x+16)(x^2-2x+2)

f(x) = x^4 -10x^3 +34x^2 -48x +32

is the most simple

polynomial with real coefficients and zeros = 4,4,and 1+i

other answers are

f(x)= ax^4 -10ax^3 +34ax^2 -48ax +32a

where a= any integer (except 0)

such as a=2

f(x) = 2x^4 -20x^2 +68x^2 -96x +64