The number of solution z3+z¯=0  is

z3+z¯=0    ⇒    z3=−  z¯ ⇒       |z|3=|−z¯|    ⇒    |z|3=|z|            |z|  (|z|−1)  (|z|+1)=0 ⇒       |z|=0   or  |z|   =   1               [∵  |z|+1>0] If        |z|=0,  then z=0 If        |z|=1  we get  |z|2=1    ⇒    z  z¯=1 Thus, z3+z¯=0    ⇒    z3+1/z=0 ⇒       z4+1=0 This equation has four non-zero and distinct roots. Therefore, the given equation has five roots.TIP It is unnecessary to find roots of z4+1=0