the common roots of the equation z^3+2z^2+2z+1=0 and z^1985 + z^100 + 1=0 are​

Dear Student,

To find the common roots of z3+2z2+2z+1=0 and z1985+z100+1=0, first let us find roots of z3+2z2+2z+1=0.
z3+2z2+2z+1=0z3+1+2zz+1=0z3+13+2zz+1=0z+1z2-z+1+2zz+1=0z+1z2-z+1+2z=0z+1z2+z+1=0z+1=0 z=-1z2+z+1=0 z=ω, ω2

Hence, roots of z3+2z2+2z+1=0 are -1, ω, ω2.
For, z=-1z1985+z100+1=-11985+-1100+1=-1+1+1=10For, z=ωz1985+z100+1=ω1985+ω100+1=ω2+ω+1=0  ω3=1 and ω2+ω+1=0For, z=ω2z1985+z100+1=ω21985+ω2100+1=ω3970+ω200+1=ω+ω2+1=0

Hence, roots of z1985+z100+1=0 are ω, ω2.
Thus, the common roots of z3+2z2+2z+1=0 and z1985+z100+1=0 are ω, ω2.
Regards,
 

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