已知a+b+c=0,求a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值

已知a+b+c=0,求a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值
已知a+b+c=0,求a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值

已知a+b+c=0,求a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值
A/B +A/C +B/C +B/A +C/A+C/B
=A+C/B +B+C/A +A+B/C
=-1+ -1+ -1 =-3

test

a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)
＝a/b＋a/c＋b/a＋b/c＋c/a＋c/b
＝(b+c)/a＋(c+a)/b＋(a+b)/c
∵a+b+c=0
→a+b=-c，c+a=-b，b+c=-a
∴原式＝-1-1-1＝-3

分别把a、b、c和括号里相乘得
a/b+a/c+b/c+b/a+c/a+c/b
把分母相同的相加得
(b+c)/a+(a+c)/b+(a+b)/c
又a+b+c=0得
b+c=-a a+c=-b a+b=-c分别带入上式得
-a/a+(-b/b)+(-c/c)=-3
a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值是-3

刚回答的朋友没有加括号,应该是
A/B +A/C +B/C +B/A +C/A+C/B
=(A+C)/B +(B+C)/A +(A+B)/C
=(-B)/B + (-A)/C + (-C)/C
=-1+ -1+ -1
=-3