因式分解(a+b)^3+(b+c)^3+(c+a)^3+a^3+b^3+c^3越快越好

Question

因式分解(a+b)^3+(b+c)^3+(c+a)^3+a^3+b^3+c^3
越快越好

Answer 1

(a+b)^3+(b+c)^3+(c+a)^3+a^3+b^3+c^3
= { (a+b)^3 + c^3 } + { a^3+ (b+c)^3 } + { (a+c)^3 + b^3 }
= (a+b+c){(a+b)^2-(a+b)c+c^2} + (a+b+c){a^2-a(b+c)+(b+c)^2} +(a+b+c){(a+c)^2-(a+c)b+b^2}
= (a+b+c){a^2+2ab+b^2-ac-bc+c^2} + (a+b+c){a^2-ab-ac+b^2+2bc+c^2} +(a+b+c){a^2+2ac+c^2-ab-bc+b^2}
= (a+b+c){a^2+2ab+b^2-ac-bc+c^2 + a^2-ab-ac+b^2+2bc+c^2 + a^2+2ac+c^2-ab-bc+b^2}
= (a+b+c){3a^2+3b^2+3c^2}
= 3 (a+b+c)(a^2+b^2+c^2)