Question

已知非零实数a,b,c满足a+b+c=0,求证(1).a^3+b^3+c^3=3abc

Answer 1

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