a3(b-c)+b3(c-a)+c3(a-b)

问题描述:

a3(b-c)+b3(c-a)+c3(a-b)

a3(b-c)+b3(c-a)+c3(a-b)
=a3b-a3c+b3c-b3a+c3a-c3b
=a3b-b3a-(a3c-b3c)+c3(a-b)
=ab(a2-b2)-c(a3-b3)+c3(a-b)
=ab(a+b)(a-b)-c(a-b)(a 2+ab+b 2)+c3(a-b)
=(a-b)[ab(a+b)-c(a 2+ab+b 2)+c3]
=(a-b)[b 2(a-c)-c(a 2-c2)+ab(a-c)]
=(a-b)(a-c)[b 2-c(a+c)+ab]
=(a-b)(a-c)[(b 2-c2)+a(b-c)]
=(a+b+c)(a-b)(b-c)(c-a).