\(a+b+c=0\\\)
\(\Rightarrow\left(a+b+c\right)^3=0\)
\(\Rightarrow a^3+b^3+c^3+3a^2b+3ab^2+3b^2c+3bc^2+3a^2c+3ac^2+6abc=0\)
=> \(a^3+b^3+c^3+\left(3a^2b+3ab^2+3abc\right)+\left(3b^2c+3bc^2+3abc\right)+\left(3a^2c+3ac^2+3abc\right)-3abc=0\)
\(\Rightarrow a^3+b^3+c^3+3ab\left(a+b+c\right)+3bc\left(a+b+c\right)+3ac\left(a+b+c\right)=3abc\)
\(Do\) \(a+b+c=0\)
\(\Rightarrow a^3+b^3+c^3=3abc\)(đpcm)
Cách này nhanh hơn nè bn!
\(a+b+c=0\Rightarrow c=-\left(a+b\right)\)
\(\Rightarrow c^3=-\left(a+b\right)^3\)
\(\Rightarrow a^3+b^3+c^3=a^3+b^3+-\left(a+b\right)^3\)
\(=a^3+b^3-a^3-b^3-3ab\left(a+b\right)\)
\(=-3ab\left(a+b\right)=-3ab.-c=3abc\)
