Question

当x+y+z=0,x的立方加y的立方加z的立方等于3xyz怎么证明

Answer 1

证明:由于x+y+z=0,故有z=-x-y

左边=x^3+y^3+z^3=x^3+y^3+(-x-y)^3

=x^3+y^3-(x+y)^3

=x^3+y^3-(x^3+3x^2y+3xy^2+y^3)

=x^3+y^3-x^3-3x^2y-3xy^2-y^3=-3x^2y-3xy^2

右边=3xyz=3xy(-x-y)=-3x^2y-3xy^2

所以有:x^3+y^3+z^3=3xyz