The equation is expressed as, z3+2z2+2z+1=0 (z+1)(z2−z+1)+2z(z+1)=0 (z+1)(z2+z+1)=0 After solving the above expression, we get z=−1 and z=ω,ω2 If, z=−1 then, z2018+z2017+1=0 (−1)2018+(−1)2017+1=0 1−1+1=1≠0 If, z = ω then, z2018+z2017+1=0 (ω)2018+(ω)2017+1=0 ω2+ω+1=0 If, z=ω2 then, z2018+z2017+1=0 (ω2)2018+(ω2)2017+1=0 ω2+ω+1=0 Hence, the common roots are ω and ω2. Consider the equation, z4+z2+1=0 If, z=ω then, ω2+ω+1=0 If, z=ω2 then, ω2+ω+1=0 Thus, the equation z4+z2+1=0, satisfy the common roots.