Question

关于分式计算的题

已知a+b+c=o,求a(1/b+1/c)+b(1/a+1/c)+c(1/b+1/a)的值
一定要有过程啊，谢谢各位大哥大姐

Answer 1

很简单：
a(1/b+1/c)+b(1/a+1/c)+c(1/b+1/a)
=a(b+c)/bc+b(a+c)/ac+c(a+b)/ab
根据a+b+c=0有b+c=-a,a+c=-b,a+b=-c
得：a(b+c)/bc+b(a+c)/ac+c(a+b)/ab
= -(a^3+b^3+c^3)/abc
= -((a+b)(a^2-ab+b^2)+c^3)/abc
=-(-c*(a^2-ab+b^2)+c^3)/abc
=-(-(a^2-ab+b^2)+c^2)/ab
=-(-(a^2-ab+b^2)+(-a-b)^2)/ab
=-(-(a^2-ab+b^2)+a^2+b^2+2ab)/ab
=-(3ab)/ab
=-3

Answer 2

a(1/b+1/c)+b(1/a+1/c)+c(1/b+1/a)
=a/b+a/c+b/a+b/c+c/b+c/a
=(a+c)/b+(a+b)/c+(b+c)/a+b/b+a/a+c/c-3
=(a+b+c)/b+(a+b+c)/c+(a+b+c)/a-3
=-3