Let z=x+iy, x,y∈R, then
z3+¯¯¯z=0⇒(x+iy)3+(x−iy)=0⇒x3+(iy)3+3ixy(x+iy)+x−iy=0⇒(x3+x−3xy2)−i(y3−3x2y+y)=0⇒x3+x−3xy2=0, y3−3x2y+y=0
So, solving
x3+x−3xy2=0⋯(1)⇒x=0 or x2−3y2+1=0y3−3x2y+y=0⋯(2)
When x=0, using equation (2), we get
y(y2+1)=0⇒y=0(∵y∈R)
When x2−3y2+1=0 using equation (2), we get
y3−3x2y+y=0⇒y(y2−3x2+1)=0⇒y(y2+1−3(3y2−1))=0⇒y(−8y2+4)=0⇒y=0, y=±1√2
But when y=0
x2+1=0, we don't get real solutions, so
y=±1√2⇒x=±1√2
Therefore,
(x,y)=(0,0), (1√2,1√2), (−1√2,−1√2), (1√2,−1√2), (−1√2,1√2)
Hence, the number of solution is 5.