If z = 2 + i, prove that `z^(3) + 3z^(2) - 9z + 8 = (1 + 14i)`.

If z = 3 + 2i , prove that z^(2) - 6z + 13 = 0 and hence deduce that 3z^(3) - 13z^(2) + 9z + 65 = 0 .

If z= -3 + sqrt2i , then prove that z^(4) + 5z^(3) + 8z^(2) + 7z + 4 is equal to -29

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that (z_(1) + z_(2)) z_(3) = z_(1) z_(3) + z_(2) z_(3)

If z_(1) = 3, z_(2) = -7i, and z_(3) = 5 + 4i, show that z_(1)(z_(2) + z_(3)) = z_(1) z_(2) + z_(1) z_(3)

If z_(1) = 1 - 3i, z_(2) = -4i and z_(3) = 5, show that (z_(1) + z_(2)) + z_(3) = z_(1) + (z_(2) + z_(3))

If z_1 = 4 - 7i , z_2 = 2 + 3i and z_3 = 1 + i show that z_1 + (z_2 + z_3) = (z_1 + z_2)+z_3

If z = -5 + 3i , find the value of (z^(4)+9z^(3)+26z^(2)-14z + 8) .

If z_(1)=3 + 4i,z_(2)= 8-15i , verify that |z_(1) + z_(2)| lt |z_(1)| + |z_(2)|

If z_(1)=3 + 4i,z_(2)= 8-15i , verify that |z_(1)z_(2) |= |z_(1)| |z_(2)|

If z_(1)=3 + 4i,z_(2)= 8-15i , verify that |z_(1) + z_(2)|^(2) + |z_(1)-z_(2)|^(2)= 2(|z_(1)|^(2) + |z_(2)|^(2))