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

Answer 1

a+b+c等于0
a+b=-c
a^2+2ab+b^2=c^2
a^2+b^2=c^2=2ab
a（1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)
=a（b+c/bc)+b(a+c/ac)+c(a+b/ab)
=a（-a/bc)+b(-b/ac)+c(-c/ab)
=-(a^2/bc+b^2/ac+c^2/ab)
=-(a^3+b^3+c^3/abc)
=-((a+b)(a^2-ab+b^2)+c^3)/abc
=-((-c)(c^2-2ab-ab)+c^3)/abc
=-((-c)(c^2-3ab)+c^3)/abc
=-(-c^3+3abc+c^3)/abc
=-3abc/abc
=-3

Answer 2

因为a+b+c=0.所以a+b=-c，b+c=-a，a+c=-b
a（1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)=-3

Answer 3

a+b+c等于0
a+b=-c
a^2+2ab+b^2=c^2
a^2+b^2=c^2=2ab
a（1/b+1/c)+b(1/c+1/a)+c(1/a+1/b)
=a（b+c/bc)+b(a+c/ac)+c(a+b/ab)
=a（-a/bc)+b(-b/ac)+c(-c/ab)
=-(a^2/bc+b^2/ac+c^2/ab)
=-(a^3+b^3+c^3/abc)
=-((a+b)(a^2-ab+b^2)+c^3)/abc
=-((-c)(c^2-2ab-ab)+c^3)/abc
=-((-c)(c^2-3ab)+c^3)/abc
=-(-c^3+3abc+c^3)/abc
=-3abc/abc
=-3

Answer 4

a^2（b+c)+b^2(c+a)+c^2(a+b)/abc=(a^2*b+b^2*a)+(a^2*c+c^2*a)+(b^2*c+c^2*b)=-3abc/abc=-3

Answer 5

a+b=-c a+c=-b b+c=-a

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