Question

UsUsi remainder theorem show that a+b, b+c, c+a are factors of (a+b+c)3-(a3+b3+c3)

Answer 1

Let f(a) = (a + b + c)³ - (a³+b³+c³)

Put a = – b in f(a),
we get,

f(– b) = ( – b + b + c)³ – [(– b )³+b³+c³]
= c³ – [– b³ + b³ + c³]
= c³ – c³
∴ f(– b) = 0

Hence (a + b) is a factor of [(a + b + c)³ - (a³+b³+c³)]

Similarly, we can prove (b + c) and (c + a) are factors of [(a + b + c)³ - (a³+b³+c³)].