The roots of the equation `z^n = (z + 3)^n`

If nin Ngt1, find the sum of real parts of the roots of the equation z^(n)=(z+1)^(n).

If nin Ngt1, find the sum of real parts of the roots of the equation z^(n)=(z+1)^(n).

Prove that none of the roots of the equation z^(n) = 2(1+ z+z^2 + …...+ z^(n-1)), n gt 1 , lies outside the circle |z|=3 .

If n>1, show that the roots of the equation z^(n)=(z+1)^(n) are collinear.

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

If n >1 , show that the roots of the equation z^n=(z+1)^n are collinear.

Prove that, for integral value of n ge1 , all the roots of the equation nz^(n) =1 + z+ z^2 +….+z^(n) lie within the circle |z|=(n)/(n-1) .

zo is one of the roots of the equation z^n cos theta0+ z^(n-1) cos theta2 +. . . . . . + z cos theta(n-1) + cos theta(n) = 2, where theta in R , then (A) |z0| 1/2 (C) |z0| = 1/2

If z_(1),z_(2),z_(3),…,z_(n-1) are the roots of the equation z^(n-1)+z^(n-2)+z^(n-3)+…+z+1=0 , where n in N, n gt 2 and omega is the cube root of unity, then