`((a-b)^(3)+(b-c)^(2)+(c-a)^(3))/(9(a-b)(b-c)(c-a))=?`

Prove that ((a^(2)-b^(2))^(3)+(b^(2)-c^(2))^(3)+(c^(2)-a^(2))^(3))/((a-b)^(3)+(b-c)^(3)+(c-a)^(3))=(a+b)(b+c)(c+a)

The expression (a-b)^3+\ (b-c)^3+\ (c-a)^3 can be factorized as (a) (a-b)(b-c)(c-a) (b) 3(a-b)(b-c)(c-a) (c) -3\ (a-b)(b-c)(c-a) (d) (a+b+c)(a^2+b^2+c^2-a b-b c-c a)

The value of [{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}/{(a-b)^3+(b-c)^3+(c-a)^3}] = (1) 3(a+b)(b+c)(c+a) (2) 3(a-b)(b-c)(c-a) (3) (a+b)(b+c)(c+a) (4) 1

The expression (a-b)^(3)+(b-c)^(3)+(c-a)^(3) can be factorized as (a)(a-b)(b-c)(c-a)(b)3(a-b)(b-c)(c-a)(c)-3(a-b)(b-c)(c-a)(d)(a+b+c)(a^(2)+b^(2)+c^(2)-ab-bc-ca)

Prove the following identities : |{:(a,a^(2),a^(3)),(b,b^(2),b^(3)),(c,c^(2),c^(3)):}|=abc(a-b)(b-c)(c-a) .

Prove that ((a)/(b)-(b)/(c))^(3)+((b)/(c)-(c)/(a))^(3)+((c)/(a)-(a)/(b))^(3)=(3(ca-b^(2))(ab-c^(2))(bc-a^(2)))/(a^(2)b^(2)c^(2))

Show that: abs((a,a^2,a^3),(b,b^2,b^3),(c,c^2,c^3))=abc(a-b)(b-c)(c-a)

Prove that a^3-(2a-b-c)^3+(b-2c)^3-(2b-c-a)^3=3(b+c-a)(a+b-2c)(2a-2b-c)

If a,b,c are in AP, than show that a^(2)(b+c)+b^(2)(c+a)+c^(2)(a+b)=(2)/(9)(a+b+c)^(3) .