Question

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

Answer 1

Answer:

Step-by-step explanation:

We are given to prove the following:

(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)=2(a^3+b^3+c^3-3abc).

We will be using the following cubic formula in the proof:

(a+b)^3=a^3+b^3+3a^2b+3ab^2.

We have

L.H.S.\\\\=(a+b)^3+(b+c)^3+(c+a)^3-3(a+b)(b+c)(c+a)\\\\=a^3+b^3+3a^2b+3ab^2+b^3+c^3+3b^2c+3bc^2+c^3+a^3+3c^2a+3ca^2-3(ab+ac+b^2+bc)(c+a)\\\\=2a^3+2b^3+2c^3+3a^2b+3ab^2+3b^2c+3bc^2+3c^2a+3ca^2-3abc-3ac^2-3b^2c-3bc^2-3a^2b-3a^2c-3ab^2-3abc\\\\=2a^3+2b^3+2c^3-6abc\\\\=2(a^3+b^3+c^3-3abc)\\\\=R.H.S.

Hence proved.

Answer 2

Step-by-step explanation:

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