Answer
(a) $x^4-2x^3+3x^2-2x+2$
(b) $x^2-(1+2i)x-1$
Work Step by Step
(a) Because the polynomial has real coefficients, by using the Conjugate Zeros Theorem
we know that the least number of zeros are 4 with $\pm i,1\pm i$, so the polynomial
with real coefficients of the smallest possible degree is
$P(x)=(x-i)(x+i)(x-1-i)(x-1+i)=(x^2+1)[(x-1)^2+1]
=(x^2+1)(x^2-2x+2)=x^4-2x^3+3x^2-2x+2$
(b) $P(x)=(x-i)(x-1-i)=x^2-x-ix-ix+i-1=x^2-(1+2i)x-1$
