Answer to Question #10684 in Algebra for nksprasad

Question

using factor theorem, show that a+b, b+c, c+a are the factors of (a+b+c)^3 - (a^3+b^3+c^3)

Answer 1

If we consider f(a,b,c)=(a+b+c)^3 - (a^3+b^3+c^3) as a f(a)=(a+b+c)^3 -
(a^3+b^3+c^3) where b,c are parameters, then values a=-b
and a=-c are roots
of equation f(a)=0. So, (a+b) and (a+c) are factors of f(a)=(a+b+c)^3 -
(a^3+b^3+c^3).
Similar consideration f(b)=(a+b+c)^3 - (a^3+b^3+c^3) where
a,c are parameters and b=-c are root of f(b)=0 gives us that (b+c) are
factor
of (a+b+c)^3 - (a^3+b^3+c^3).