非0实数a b c满足a+b+c=0,求证a³+b³+c³=3abc

题目

非0实数a b c满足a+b+c=0,求证a³+b³+c³=3abc

答案

a+b=-c (a+b)^2=c^2 a^2+b^2+2ab=c^2 a^2+b^2=c^2-2ab a^3+b^3+c^3 =(a+b)(a^2-ab+b^2)+c^3 =(-c)(a^2-ab+b^2)+c^3 =c(c^2-a^2+ab-b^2) =c[c^2-(a^2+b^2)+ab] =c(c^2-c^2+2ab+ab) =c*3ab =3abc