(vi) z3+z2+z+1=0z^3 + z^2 + z + 1 = 0z3+z2+z+1=0 Solution Group terms:(z3+z2)+(z+1)=z2(z+1)+1(z+1)Factor out (z+1):=(z+1)(z2+1)=0Apply zero product property:z+1=0 ⟹ z=−1z2+1=0 ⟹ z2=−1 ⟹ z=±iTherefore: z=−1,i,−i\begin{aligned} & \boxed{\text{Group terms:}} \\ \\ & (z^3 + z^2) + (z + 1) = z^2(z + 1) + 1(z + 1) \\ \\ & \boxed{\text{Factor out } (z + 1):} \\ \\ & = (z + 1)(z^2 + 1) = 0 \\ \\ & \boxed{\text{Apply zero product property:}} \\ \\ & z + 1 = 0 \implies z = -1 \\ \\ & z^2 + 1 = 0 \implies z^2 = -1 \implies z = \pm i \\ \\ & \boxed{\text{Therefore: } z = -1, i, -i} \end{aligned}Group terms:(z3+z2)+(z+1)=z2(z+1)+1(z+1)Factor out (z+1):=(z+1)(z2+1)=0Apply zero product property:z+1=0⟹z=−1z2+1=0⟹z2=−1⟹z=±iTherefore: z=−1,i,−i