Prove that `|[1,1,1],[a,b,c],[a^3,b^3,c^3]|=(a-b)(b-c)(c-a)(a+b+c)`

Using the property of determinants and without expanding prove that abs([1,1,1],[a,b,c],[a^3,b^3,c^3])=(a-b)(b-c)(c-a)(a+b+c)

1,1,1a,b,ca^(3),b^(3),c^(3)]|=(a-b)(b-c)(c-a)(a+b+c)

Prove that |[1,a,a^3],[1,b,b^3],[1,c,c^3]|=(a-b)(b-c)(c-a)(a+b+c)

Prove that: (i) |{:(,1,1,1),(,a,b,c),(,a^(3),b^(3),c^(3)):}|=(a-b)(b-c)(c-a)(a+b+c)

If a,b,are distinct,show that [[1,1,1a,b,ca^(3),b^(3),c^(3)]]=(b-c)*(c-a)*(a-b)(a+b+c)

The value of the determinant /_\=|(1,1,1),(a,b,c),(a^3,b^3,c^3)| is (A) (a-b)(b-c)(c-a)(a+b+c) (B) abc(a+b)(b+C)(c+a) (C) (a-b)(b-c)(c-a) (D) none of these

Prove that |(1,a,a^2),(1,b,b^2),(1,c,c^2)|=(a-b)(b-c)(c-a)

Prove that =|1 1 1a b c b c+a^2a c+b^2a b+c^2|=2(a-b)(b-c)(c-a)

Show that det[[1,1,1a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)det[[a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)det[[a^(2),b^(2),c^(2)a^(3),b^(3),c^(3)]]=(b-c)(c-a)(a-b)(bc+ca+ab)

Prove that (a+b+c)^(3)-a^(3)-b^(3)-c^(3)=3(a+b)(b+c)(c+a) .