prove: `(a+b+c)(a+bw + cw^2) (a+bw^2 + cw) = a^3 + b^3 + c^3 - 3abc` w is cube root of unity

prove: (a+b+c)(a+bw+cw^(2))(a+bw^(2)+cw)=a^(3)+b^(3)+c^(3)-3abcw is cube root of unity

If a , b , c are real numbers, prove that \begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \\ \end{vmatrix}= -(a+b+c)(c+b w+c w^2)(a+ bw^2+cw), where \ \ w is a complex cube root of unity.

If x=a+bw+cw^2 ,show that x^3-3ax^2+3(a^2-bc)x=a^3+b^3+c^3-3abc .when w be the an imaginarry cube root of unity.

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.

(a+b) (a omega+b omega^(2)) (aw^(2)+bw) =?

If a , b , c are real numbers, prove that |[a,b,c],[b,c,a],[c,a,b]|=-(a+b+c)(c+b w+c w^2)(a+b w^2+c w), where w is a complex cube root of unity.

Using properties of determinant, If f(x)= a + bx + cx^(2) , prove that |(a,b,c),(b,c,a),(c,a,b)|= -f(1)f(omega)f(omega^(2)), omega is a cube root of unity

If x= a+ b, y = a omega + b omega^(2), z= a omega^(2) + b omega , then the value of x^(3) + y^(3) + z^(3) is equal to (where omega is imaginary cube root of unity)

If x=a+b,y=aw+bw^(2) and z=aw^(2)+bw, where w is an imaginary cube. root of unity,prove that x^(2)+y^(2)+z^(2)=6ab