Simplify `(a+b+c)(a+bomega+comega^2)(a+bomega^2+comega)`

Prove that a^3 + b^3 + c^3 – 3abc = (a + b + c) (a + bomega + comega^2) (a + bomega^2 + "c"omega) , where omega is an imaginary cube root of unity.

omega is an imaginary root of unity. Prove that (i) ( a + bomega + comega^(2))^(3) + (a+bomega^(2) + comega)^(3) = (2a-b-c)(2b -a -c)(2c -a-b) (ii) If a+b+c = 0 then prove that (a + bomega + comega^(2))^(3)+(a+bomega^(2) + comega)^(3) = 27abc .

Let omega be the imaginary cube root of unity and (a+bomega + comega^2)^(2015) =(a+bomega^2 + c omega) where a,b,c are unequal real numbers . Then the value of a^2+b^2+c^2-ab-bc-ca equals.

If a,b,c are distinct integers and omega(ne 1) is a cube root of unity, then the minimum value of |a+bomega+comega^(2)|+|a+bomega^(2)+comega| is

Simplify : (a) (a+b+c)(a-b-c)

Simplify (ab- c) ^(2) + 2 abc

If omega be an imaginary cube root of unity, show that (a+bomega+comega^2)/(aomega+bomega^2+c) = omega^2

Statement-1: If a,b,c are distinct real number and omega( ne 1) is a cube root of unity, then |(a+bomega+comega^(2))/(aomega^(2)+b+comega)|=1 Statement-2: For any non-zero complex number z,|z / bar z)|=1