#)Giải :
a) \(x+y+z=0\Leftrightarrow x+y=-z\Leftrightarrow\left(x+y\right)^3=\left(-z\right)^3\Leftrightarrow x^3+3x^2y+3xy^2+y^3=\left(-z\right)^3\)
\(\Leftrightarrow x^3+y^3+z^3=-3x^2y-3xy^2\Leftrightarrow x^3+y^3+z^3=-3xy\left(-z\right)\) hay 3xyz (đpcm)
b) \(x=\left(a-b\right)^3+\left(b-c\right)^3+\left(c-a\right)^3\)
\(\Leftrightarrow a^3+b^3=\left(a+b\right)^3-3ab\left(a+b\right)\) (Áp dụng hằng đẳng thức)
\(\Leftrightarrow x=\left[\left(b-c\right)^3+\left(c-a\right)^3\right]+\left(a-b\right)^3\)
\(=\left[\left(b-a\right)^3+\left(c-a\right)^3\right]-3\left(b-c\right)\left(c-a\right)\left[\left(b-c\right)+\left(c-a\right)\right]+\left(a-b\right)^3\)
\(=\left(b-a\right)^3-3\left(b-c\right)\left(c-a\right)\left(b-a\right)+\left(a-b\right)^3\)
\(=\left[-\left(a-b\right)^3\right]-3\left(b-c\right)\left(c-a\right)\left[-\left(a-b\right)\right]+\left(a-b\right)^3\)
\(=-\left(a-b\right)^3+3\left(a-b\right)\left(b-c\right)\left(c-a\right)+\left(a-b\right)^3=3\left(a-b\right)\left(b-c\right)\left(c-a\right)\)
