(iii) 5z5−5z=05z^5 - 5z = 05z5−5z=0 Solution Factor out common factor 5z:5z(z4−1)=0Apply zero product property:5z=0 ⟹ z=0z4−1=0 ⟹ z4=1Difference of squares twice:(z2−1)(z2+1)=0z2−1=0 ⟹ z2=1 ⟹ z=±1z2+1=0 ⟹ z2=−1 ⟹ z=±iTherefore: z=0,1,−1,i,−i\begin{aligned} & \boxed{\text{Factor out common factor 5z:}} \\ \\ & 5z(z^4 - 1) = 0 \\ \\ & \boxed{\text{Apply zero product property:}} \\ \\ & 5z = 0 \implies z = 0 \\ \\ & z^4 - 1 = 0 \implies z^4 = 1 \\ \\ & \boxed{\text{Difference of squares twice:}} \\ \\ & (z^2 - 1)(z^2 + 1) = 0 \\ \\ & z^2 - 1 = 0 \implies z^2 = 1 \implies z = \pm 1 \\ \\ & z^2 + 1 = 0 \implies z^2 = -1 \implies z = \pm i \\ \\ & \boxed{\text{Therefore: } z = 0, 1, -1, i, -i} \end{aligned}Factor out common factor 5z:5z(z4−1)=0Apply zero product property:5z=0⟹z=0z4−1=0⟹z4=1Difference of squares twice:(z2−1)(z2+1)=0z2−1=0⟹z2=1⟹z=±1z2+1=0⟹z2=−1⟹z=±iTherefore: z=0,1,−1,i,−i