Question

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

Answer 1

let x=a^2-b^2
y=b^2-c^2
z=c^2-a^2
now,
x+y+z=a^2-b^2+b^2-c^2+c^2-a^2=0
we know that if x+y+z=0
so,x^3+y^3+z^3=3xyz
(From identity 10)
(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3=3(a^2-b^2)(b^2-c^2)(c^2-a^2)=3(a+b)(b+c)(c+a)(a-b)(b-c)(c-a)