Question

已知abc不等于0,且a+b+c=0,求

已知abc不等于0,且a+b+c=0,求代数式(a^2)/（bc）+(b^2)/(ac)+(c^2)/ab的值

Answer 1

(a^2)/（bc）+(b^2)/(ac)+(c^2)/ab
=(a^3)/（abc）+(b^3)/(abc)+(c^3)/abc (分子分母扩大相同倍数,大小不变）
=[a^3+b^3+c^3]/abc(接下来分解因式即可）
=[( a+b)^3-3a^2b-3ab^2]+c^3/abc
=[(a+b)^3+c^3]-(3a^2b+3ab^2) /abc
=[(a+b+c)[(a+b)^2-(a+b)c+c^2]-3ab(a+b)]/abc
=[(a+b+c)(a^2+b^2+2ab-ac-bc+c^2)-3ab(a+b)]/abc
=[(a+b+c)(a^2+b^2+c^2-ab-ac-bc)+3abc]/abc
=[0+3abc]/abc
=3