Verify the following: (a+b+c) (a^2+b^2+c^2-a b-b c-c a )=a^3+b^3+c^3-3 a b c

(a+b+c) (a^2+b^2+c^2-a b-b c-c a )=a^3+b^3+c^3-3 a b c Taking LHS (a+b+c) (a^2+b^2+c^2-a b-b c-c a ) =a (a^2+b^2+c^2-a b-b c-c a )+b (a^2+b^2+c^2-a b-b c-c a ) c (a^2+b^2+c^2-a b-b c-c a ) =a^3+a b^2+a c^2-a^2 b-a b c-a^2 c+b a^2+b^3+b c^2-b^2 a-b^2 c-b c a+c a^2+c b^2+c^3-c a b-c^2 b-c^2 a =a^3+b^3+c^3-3 a b c Hence verified

Verify the following: (a+b+c) (a^2+b^2+c^2-a b-b c-c a )=a^3+b^3+…

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Question

Verify the following:
$(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)=a^3+b^3+c^3-3 a b c$

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Answer

$(a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right)=a^3+b^3+c^3-3 a b c$
Taking $LHS$
$ (a+b+c)\left(a^2+b^2+c^2-a b-b c-c a\right) $
$ =a\left(a^2+b^2+c^2-a b-b c-c a\right)+b\left(a^2+b^2+c^2-a b-b c-c a\right) c\left(a^2+b^2+c^2-a b-b c-c a\right) $
$ =a^3+a b^2+a c^2-a^2 b-a b c-a^2 c+b a^2+b^3+b c^2-b^2 a-b^2 c-b c a+c a^2+c b^2+c^3-c a b-c^2 b-c^2 a $
$ =a^3+b^3+c^3-3 a b c $
Hence verified

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