1;\(A=x^3+y^3+z^3-3xyz\)

\(A=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)

\(A=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)

\(A=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)=0\)

2;Nếu A = 0

Điều ngược lại đúng khi x^2+y^2+z^2-xy-yz-xz khác 0

Ta đi chứng minh A phụ thuộc vào x+y+z

\(A=x^3+y^3+z^3-3xyz.\)

\(=\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\)

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-z\left(x+y\right)+z^2\right]-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)

\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

Mà x^2+y^2+z^2-xy-yz-xz>0

nên  x+y+z =0 thì A=0

\(A=x^3+y^3+z^3-3xyz=\left(x+y\right)^3+z^3-3xyz-3x^2y-3y^2x=\left(x+y+z\right)\left(x^2+2xy-xz-zy+z^2+y^2\right)-3xy\left(x+y+z\right)=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-zx\right)=0\)

\(xet:x=y=z=1\Rightarrow A=1+1+1-3=0\Rightarrow dieunguoclaichuachacdadung\)

Lời giải:

Ta có:

$A=x^3+y^3+z^3-3xyz=(x+y)^3-3xy(x+y)+z^3-3xyz$

$=(x+y)^3+z^3-3xy(x+y)-3xyz$

$=(x+y+z)[(x+y)^2-z(x+y)+z^2]-3xyz(x+y+z)$

$=(x+y+z)[(x+y)^2-z(x+y)+z^2-3xyz]$

$=(x+y+z)(x^2+y^2+z^2-xy-yz-xz)$
a)

Nếu $x+y+z=0\Rightarrow A=0.(x^2+y^2+z^2-xy-yz-xz)=0$

b)

Nếu $A=0\Leftrightarrow (x+y+z)(x^2+y^2+z^2-xy-yz-xz)=0$

$\Leftrightarrow x+y+z=0$ hoặc $x^2+y^2+z^2-xy-yz-xz=0$

Nếu $x+y+z=0$ thì điều ngược lại đúng

Nếu $x^2+y^2+z^2-xy-yz-xz=0$ thì $x=y=z$ (như bài trước đã CM). Như vậy trường hợp này không đủ cơ sở kết luận $x+y+z=0$

Tổng hợp 2 TH lại thì khi $A=0$ thì chưa chắc $x+y+z=0$, tức là điều ngược lại không đúng.

\(x^3+y^3+z^3-3xyz=\left(x+y\right)^3+z^3-3xyz-3x^2y-3xy^2\)

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)

\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2-3xy\right]\)

\(=0\)

\(\Rightarrow x^3+y^3+z^3=3xyz\)

\(a,\left(x+y+z\right)^3-x^3-y^3-z^3\\ =\left[\left(x+y\right)+z\right]^3-x^3-y^3-z^3\\ =\left(x+y\right)^3+z^3+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\\ =x^3+y^3+z^3+3xy\left(x+y\right)+3z\left(x+y\right)\left(x+y+z\right)-x^3-y^3-z^3\\ =\left(x+y\right)\left(3xy+3xz+3yz+3z^2\right)\\ =3\left(x+y\right)\left[x\left(y+z\right)+z\left(y+z\right)\right]\\ =3\left(x+y\right)\left(y+z\right)\left(x+z\right)\)

\(b,x^3+y^3+z^3-3xyz\\ =\left(x+y\right)^3-3xy\left(x+y\right)+z^3-3xyz\\ =\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2\right)-3xy\left(x+y+z\right)\\ =\left(x+y+z\right)\left(x^2+y^2+z^2-xz-yz+2xy-3xy\right)\\ =0\left(x^2+y^2+z^2-xz-yz-xy\right)=0\\ \Leftrightarrow x^3+y^3+z^3=3xyz\)

Ta có: \(x^3+y^3+z^3=3xyz\)

\(\Rightarrow x^3+y^3+z^3-3xyz=0\)

\(\Rightarrow x^3+y^3+z^3-3xyz=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)

\(\Rightarrow\left(x+y\right)^3=\left(-z\right)^3\)

\(\Rightarrow x+y=-z\)\(\Rightarrow x+y+z=0\left(đpcm\right)\)( P/s cx ko chắc lắm :P )

That's very easy 

\(x^3+y^3+z^3=3xyz\)

\(\Leftrightarrow x^3+y^3+z^3-3xyz=0\)

\(\Leftrightarrow\left(x^3+y^3+3x^2y+3y^2x\right)+z^3-3xy\left(x+y+z\right)=0\)

\(\Leftrightarrow\left(x+y\right)^3+z^3-3xy\left(x+y+z\right)=0\)

\(\Leftrightarrow\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)=0\)

\(\Leftrightarrow\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2-3xy\right]=0\)

\(\Leftrightarrow\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x+y+z=0\left(1\right)\\x^2+y^2+z^2-xy-yz-xz=0\end{cases}}\)

Lại có : \(x^2+y^2+z^2-xy-yz-xz=0\)

Nhân 2 lên , nhóm vào ta được các cặp số : \(\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2=0\left(2\right)\)( làm tắt ) 

Do \(\hept{\begin{cases}\left(x-y\right)^2\ge0\forall x;y\\\left(y-z\right)^2\ge0\forall y;z\\\left(x-z\right)^2\ge0\forall x;z\end{cases}}\)

\(\Rightarrow\left(x-y\right)^2+\left(y-z\right)^2+\left(x-z\right)^2\ge0\forall x;y;z\left(3\right)\)

Từ ( 2 ) ; ( 3 ) \(\Rightarrow\hept{\begin{cases}x-y=0\\y-z=0\\x-z=0\end{cases}\Rightarrow x=y=z}\left(4\right)\)

Từ (1) ; (4) => đpcm

Bổ sung : \(\left(x-z\right)^2\ge0\forall x;z\)

nếu x+y+z=0 thì x^3+y^3+z^3=3xyz

Ta có : x + y + z = 0 => x + y = -z => (x + y)³ = (-z)³

=> x³ + 3x²y + 3xy² + y³ = (-z)³

=> x³ + y³ + z³ + 3x²y + 3xy² = 0

=> x³ + y³ + z³ + 3xy(x + y) = 0

Mà x + y = -z

Nên :  x³ + y³ + z³ + 3xy(-z) = 0

=>  x³ + y³ + z³ - 3xyz = 0

=> A = 0

Vậy x + y + z = 0 thì A = 0 (đpcm)

Từ:

 x + y + z = 0 

=> x + y = -z 
<=> (x + y)^3 = (-z)^3 
<=> x^3 + 3x^2y + 3xy^2 + y^3 = -z^3 
<=> x^3 + y^3 + z^3 = -3x^2y - 3xy^2 
<=> x^3 + y^3 + z^3 = -3xy(x+y) 
<=> x^3 + y^3 + z^3 = -3xy(-z) 
<=> x^3 + y^3 + z^3 = 3xyz 

P/s: Tham khảo nha

\(x^3+y^3+z^3=3xyz\)

\(\Leftrightarrow x^3+y^3+z^3=-3xy\left(-z\right)\)

\(\Leftrightarrow x^3+y^3+z^3=-3\left(x+y\right)\)

\(\Leftrightarrow x^3+y^3+z^3=-3x^2y-3xy^2\)

\(\Leftrightarrow x^3+3x^2y+3xy^2+y^3=-z^3\)

\(\Leftrightarrow\left(x+y\right)^3=\left(-z\right)^3\)

\(\Leftrightarrow x+y=-z\)

\(\Leftrightarrow x+y+z=0\)