Prove : (a+b+c)^3 - a^3-b^3-c^3=(a+b)(b+c)(c+a)
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(a+b+c)^3-a^3-b^3-c^3 ={(a+b)+c}^3- a^3-b^3-c^3 =(a+b)^3+c^3+3c(a+b)(a+b+c)-a^3-b^3-c^3 =a^3+b^3+c^3+3ab(a+b)+3c(a+b)(a+b+c)-a^3-b^3-c^3=3ab(a+b)+3c(a+b)(a+b+c)=3(a+b){ab+c(a+b+c)}=3(a+b){ab+ac+bc+c^3}=3(a+b){a(b+c)+c(b+c)} =3(a+b)(b+c)(c+a)
Hence proved
(a+b+c)^3-a^3-b^3-c^3 ={(a+b)+c}^3- a^3-b^3-c^3 =(a+b)^3+c^3+3c(a+b)(a+b+c)-a^3-b^3-c^3 =a^3+b^3+c^3+3ab(a+b)+3c(a+b)(a+b+c)-a^3-b^3-c^3=3ab(a+b)+3c(a+b)(a+b+c)=3(a+b){ab+c(a+b+c)}=3(a+b){ab+ac+bc+c^3}=3(a+b){a(b+c)+c(b+c)} =3(a+b)(b+c)(c+a)
Hence proved
tuli7:
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