a, A=ab(a−b)+bc(b−c)+ca(c−a)
=a2b−ab2+b2c−bc2+ac2−a2c
=(a2b−abc)−(ab2−b2c)+(ac2−bc2)−(a2c−abc)
=ab(a−c)−b2(a−c)+c2(a−b)−ac(a−b)
=b(a−b)(a−c)+c(c−a)(a−b)
=b(a−b)(a−c)−c(a−c)(a−b)
=(a−b)(b−c)(a−c)
b, B=a(b2−c2)+b(c2−a2)+c(a2−b2)
=ab2−ac2+bc2−a2b+a2c−b2c
=(ab2−abc)+(abc−ac2)−(b2c−bc2)−(a2b−a2c)
=ab(b−c)+ac(b−c)−bc(b−c)−a2(b−c)
=(ab+ac−bc−a2)(b−c)
=[(ab−bc)+(ac−a2)](b−c)
=[b(a−c)+a(c−a)](b−c)
=(b−a)(a−c)(b−c)
c, C=(a+b+c)3−a3−b3−c3
=a3+b3+c3+3ab(a+b)+3ac(a+c)+3bc(b+c)+6abc−a3−b3−c3
=3(a2b+ab2+a2c+ac2+b2c+bc2+2abc)
=3[ab(a+b)+bc(a+b)+c2(a+b)+ac(a+b)]
=3(a+b)(ab+bc+c2+ac)
=3(a+b)[b(a+c)+c(a+c)]=3(a+b)(a+c)(b+c)