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b, \(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)
\(=\left(x-y\right)^2\left(x-y\right)-\left(y-z\right)^2\left[\left(x-y\right)+\left(z-x\right)\right]+\left(z-x\right)^2\left(z-x\right)\)
\(=\left(x-y\right)^2\left(x-y\right)-\left(y-z\right)^2\left(x-y\right)-\left(y-z\right)^2\left(z-x\right)+\left(z-x\right)^2\left(z-x\right)\)
\(=\left(x-y\right)\left[\left(x-y\right)^2-\left(y-z\right)^2\right]-\left(z-x\right)\left[\left(y-z\right)^2-\left(z-x\right)^2\right]\)
\(=\left(x-y\right)\left(x-y-y+z\right)\left(x-y+y-z\right)-\left(z-x\right)\left(y-z-z+x\right)\left(y-z+z-x\right)\)
\(=\left(x-y\right)\left(x-2y+z\right)\left(x-z\right)-\left(z-x\right)\left(y-2z+x\right)\left(y-x\right)\)
\(=\left(x-y\right)\left(x-2y+z\right)\left(x-z\right)-\left(x-z\right)\left(y-2z+x\right)\left(x-y\right)\)
\(=\left(x-y\right)\left(x-z\right)\left(x-2y+z-y+2z-x\right)\)
\(=\left(x-y\right)\left(x-z\right)\left(3z-3y\right)\)
\(=3\left(x-y\right)\left(x-z\right)\left(z-y\right)\)
c, \(x^2y^2\left(y-x\right)+y^2z^2\left(z-y\right)-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left[\left(y-x\right)-\left(z-x\right)\right]-z^2x^2\left(z-x\right)\)
\(=x^2y^2\left(y-x\right)-y^2z^2\left(y-x\right)+y^2z^2\left(z-x\right)-z^2x^2\left(z-x\right)\)
\(=\left(x^2y^2-y^2z^2\right)\left(y-x\right)+\left(y^2z^2-z^2x^2\right)\left(z-x\right)\)
\(=y^2\left(x-z\right)\left(x+z\right)\left(y-x\right)+z^2\left(y-x\right)\left(x+y\right)\left(z-x\right)\)
\(=y^2\left(x-z\right)\left(x+z\right)\left(y-x\right)-z^2\left(y-x\right)\left(x+y\right)\left(x-z\right)\)
\(=\left(x-z\right)\left(y-x\right)\left[y^2\left(x+z\right)-z^2\left(x+y\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left(y^2x+y^2z-z^2x-z^2y\right)\)
\(=\left(x-z\right)\left(y-x\right)\left[x\left(y^2-z^2\right)+yz\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left[x\left(y-z\right)\left(y+z\right)+yz\left(y-z\right)\right]\)
\(=\left(x-z\right)\left(y-x\right)\left(y-z\right)\left(xy+xz+yz\right)\)
d, \(x^3+y^3+z^3-3xyz\)
\(=\left(x+y\right)^3+z^3-3xyz-3xy\left(x+y\right)\)
\(=\left(x+y+z\right)\left[\left(x+y\right)^2-\left(x+y\right)z+z^2\right]-3xy\left(x+y+z\right)\)
\(=\left(x+y+z\right)\left(x^2+2xy+y^2-xz-yz+z^2-3xy\right)\)
\(=\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)\)
Lời giải:
Ta có:
\(x^3(y-z)+z^3(x-y)=y^3(x-z)=y^3[(y-z)+(x-y)]\)
\(\Leftrightarrow x^3(y-z)+z^3(x-y)-y^3(y-z)-y^3(x-y)=0\)
\(\Leftrightarrow (x^3-y^3)(y-z)-(y^3-z^3)(x-y)=0\)
\(\Leftrightarrow (x-y)(x^2+xy+y^2)(y-z)-(y-z)(y^2+yz+z^2)(x-y)=0\)
\(\Leftrightarrow (x-y)(y-z)(x^2+xy+y^2-y^2-yz-z^2)=0\)
\(\Leftrightarrow (x-y)(y-z)(x^2+xy-z^2-yz)=0\)
\(\Leftrightarrow (x-y)(y-z)(x-z)(x+y+z)=0\)
Vì $x,y,z$ đôi một khác nhau nên \((x-y)(y-z)(x-z)\neq 0\). Do đó $x+y+z=0$
Khi đó:
\(x^3+y^3+z^3=(x+y)^3-3xy(x+y)+z^3\)
\(=(-z)^3-3xy(-z)+z^3=-z^3+3xyz+z^3=3xyz\)
Ta có đpcm.
Ta có : \(\left(x-y+z\right)^2=x^2-y^2+z^2\)
\(\Leftrightarrow x^2+y^2+z^2-2xy+2xz-2yz=x^2-y^2+z^2\)
\(\Leftrightarrow2y^2-2xy+2xz-2yz=0\)
\(\Leftrightarrow2y\left(y-z\right)-2x\left(y-z\right)=0\)
\(\Leftrightarrow2\left(y-x\right)\left(y-z\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}y=x\\y=z\end{matrix}\right.\)
Với x = y \(\Rightarrow\left(x-y+z\right)^n=z^n;x^n-y^n+z^n=z^n\)
\(\Rightarrow\left(x-y+z\right)^n=x^n-y^n+z^n\) ( 1 )
Với y = z \(\Rightarrow\left(x-y+z\right)^n=x^n;x^n-y^n+z^n=x^n\)
\(\Rightarrow\left(x-y+z\right)^n=x^n-y^n+z^n\) ( 2 )
Từ ( 1 ) ; ( 2 ) => ĐPCM