Question

9. Using factor theorem, show that a + b, b + c
and c + a are the factors of (a + b + c)^3-(a^3 + b^3 + c^3).​

Answer 1

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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)].