The remainder theorem (and the closely related factor theorem) gives us an immediate connection between the linear factors of a polynomial and its roots. Namely, if a E C is a root of the polynomial p(z) (i.e. p(a) = 0 ) then p(z) has (z a) as a factor. Let's answer a few questions to test our understanding of these ideas. ) = z 4z + 3 then p(1) p(z). i) If p(z) a factor of a factor of ii) If p(z) = z (1 + i)z + i then p(i) p(z). iii) If p(z) = z2 not a factor of = Number Number which shows that (z-1) which shows that (z-i) - 2x + 1 then p(-2) = Number p(z). iv) If we divide p(z) = z 4z + 4 by z 3, we get a remainder of Number v) If we divide p(x) = z 4z + 4 by z 2, we get a remainder of vi) If we divide p(z) = z 4z + 4 by z + 1, we get a remainder of which shows that (z+2) Number Number is is is