实数a,b,c都不为0,且a+b+c=0,则a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值是几?

Question

实数a,b,c都不为0,且a+b+c=0,则a(1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)的值是几?

Answer 1

解:
a+b+c=0
所以
a+b=-c
a+c=-b
b+c=-a
所以
a*(1/b+1/c)+b*(1/c+1/a)+c*(1/a+1/b)
=a/b+a/c+b/c+b/a+c/a+c/b
=(b+c)/a+(a+c)/b+(a+b)/c
=(-a)/a+(-b)/b+(-c)/c
=-1-1-1
=-3