Find the polynomial

Question

Find the polynomial. Degree 4; only zeros are 4, 3+i, and 3-i

Answer 1

x = 3 + i

x-3 = i

x^2 -6x + 9 = -1

x^2 -6x +10 = 0

(x-4)^2(x^2-6x+10) = 0

(x^2 -8x +16)(x^2 -6x +10) = 0

x^4 -14x^3 +74x^2 -176x +160 = 0

is a 4th degree polynomial with just 3 zeros, the two imaginary zeros come in a conjugate pair. The x=4 zero has multiplicity two. That's why (x-4)^2 has the exponent 2.