complex numbers polynomial

the question is "If z = 1+i is a zero of the polynomial z^3 + az^2 + bz + 10 - 6i, find the constants 'a' and 'b' given that they are real."

now becuase 1+i is a solution i am going to assume 1-i is a solution becuase the polynomial has real co-efficients. so (z - 1 - i)(z - 1 + i) = z^2 - 2z + 2 as a quadratic factor

im suspecting there could be a problem with the question becuase on my CAS calculator 1+i is certainly a solution but 1-i is NOT (substituting the answers from the back of the book a=6, b=-8)