(a+b)3+(b+c)3+(c+d)3+(d+a)3+(a+c)3+(b+d)3分解因式

原式=[(a+b)3+(c+d)3]+[(b+c)3+(d+a)3]+[(a+c)3+(b+d)3]
=(a+b+c+d)[(a+b)2-(a+b)(c+d)+(c+d)2]+(b+c+d+a)[(b+c)2-(b+c)(d+a)+(d+a)2]+(a+c+b+d)[(a+c)2-(a+c)(b+d)+(b+d)2]
=(a+b+c+d)(3a2+3b2+3c2+3d2)
=3(a+b+c+d)(a2+b2+c2+d2)
注立方和公式：a3+b3=(a+b)(a2-ab+b2)

。。

(a+b)3+(b+c)3+(c+d)3+(d+a)3+(a+c)3+(b+d)3
=(a^3+3a^2b+3ab^2+b^3)+(b^3+3b^2c+3bc^2+c^3)+(c^3+3c^2d+3cd^2+c^3)+(a^3+3a^2d+3ad^2+d^3)+(a^3+3a^2c+3ac^2+c^3)+(b^3+3b^2d+3bd^2+d^3)
=3a^3+3b^3+3c^3+3d^3+3a^2b+3a^2d+3a^2c+3b^2c+3ab^2+3b^2d+3bc^2+3ac^2+3c^2d+3cd^2+3ad^2+3bd^2
=3*[a^2*a+b^2*b+c^2*c+d^2*d+a^2(b+c+d)+b^2(a+c+d)+c^2(a+b+d)+d^2(a+b+c)]
=3(a^2+b^2+c^2+d^2)(a+b+c+d)

(a+b)3+(b+c)3+(c+d)3+(d+a)3+(a+c)3+(b+d)3
=(a+b+c+d)[(a+b)²-(a+b)(c+d)+(c+d)²]+(b+c+a+d)[(b+c)²-(b+c)(a+d)+(a+d)²]+(a+c+b+d)[(a+c)²-(a+c)(b+d)+(b+d)²]
=(a+b+c+d)[(a+b)²-(a+b)(c+d)+(c+d)²+(b+c)²-(b+c)(a+d)+(a+d)²+(a+c)²-(a+c)(b+d)+(b+d)²]
=(a+b+c+d)(a²+2ab+b²-ac-bc-ad-bd+c²+2cd+d²+b²+2bc+c²-ab-ac-bd-cd+a²+2ad+d²+a²+2ac+c²+b²+2bd+d²-ab-bc-cd-ad)
=(a+b+c+d)(3a²+3b²+3c²+3d²)
=3(a+b+c+d)(a²+b²+c²+d²)

(a+b)^3 + (b+c)^3 + (c+d)^3 + (d+a)^3 + (a+c)^3 + (b+d)^3
= [ (a+b)^3 + (c+d)^3 ] + [ (b+c)^3 + (d+a)^3 ] + [ (a+c)^3 + (b+d)^3 ]
= (a+b+c+d)[ (a+b)" - (a+b)(c+d) + (c+d)"] + (a+b+c+d)[ (b+c)" - (b+c)(d+a) + (a+d)"]
+ (a+b+c+d)[ (a+c)" - (a+c)(b+d) + (b+d)"]
= (a+b+c+d)[ (a+b)" +(c+d)" +(b+c)" +(a+d)" +(a+c)" +(b+d)" -(a+b)(c+d) -(b+c)(d+a) -(a+c)(b+d) ]
= (a+b+c+d)[ (a"+b"+2ab) +(a"+c"+2ac) +(a"+d"+2ad) +(b"+c"+2bc) +(b"+d"+2bd) +(c"+d"+2cd) -(ac+bc+ad+bd) -(ab+ac+bd+cd) -(ab+bc+ad+cd) ]
= 3(a +b +c +d)(a" +b" +c" +d")