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\(\frac{yz\left(z-y\right)+xz\left(x-z\right)+xy\left(y-x\right)}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)\(=\frac{yz\left(z-y\right)+xz\left(x-z\right)-xy\left[\left(z-y\right)+\left(x-z\right)\right]}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}\)
\(=\frac{y\left(z-y\right)\left(z-x\right)+x\left(x-z\right)\left(z-y\right)}{xyz\left(x-y\right)\left(y-z\right)\left(z-x\right)}=\frac{\left(z-y\right)\left(z-x\right)\left(y-x\right)}{xyz\left(z-y\right)\left(z-x\right)\left(y-x\right)}=\frac{1}{xyz}\)
\(A=\frac{x^2}{\left(x-y\right)\left(x-z\right)}+\frac{y^2}{\left(y-x\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)
\(=\frac{x^2}{\left(x-y\right)\left(x-z\right)}-\frac{y^2}{\left(x-y\right)\left(y-z\right)}+\frac{z^2}{\left(x-z\right)\left(y-z\right)}\)
\(=\frac{x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z-xy^2+y^2z+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left[xy-zx-zy+z^2\right]\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
Vậy A = 1
Ta có: \(\frac{x^2-yz}{\left(x+y\right)\left(x+z\right)}=\frac{x^2+xy-xy-yz}{\left(x+y\right)\left(x+z\right)}\)
\(=\frac{x\left(x+y\right)-y\left(x+z\right)}{\left(x+y\right)\left(x+z\right)}\)
\(=\frac{x}{x+z}-\frac{y}{x+y}\)
Tương tự: \(\frac{y^2-xz}{\left(x+y\right)\left(y+z\right)}=\frac{y}{y+z}-\frac{y}{x+y}\)
\(\frac{z^2-xz}{\left(x+z\right)\left(y+z\right)}=\frac{z}{y+z}-\frac{x}{x+z}\)
Do đó: \(A=\frac{x}{x+z}-\frac{y}{x+y}+\frac{y}{y+z}-\frac{x}{x+y}+\frac{z}{y+z}-\frac{x}{x+z}=0\)
\(\frac{y-z}{\left(x-y\right)\left(x-z\right)}=\frac{\left(x-z\right)-\left(x-y\right)}{\left(x-y\right)\left(x-z\right)}=\frac{1}{x-y}-\frac{1}{x-z}\)
\(\frac{z-x}{\left(y-z\right)\left(y-x\right)}=\frac{\left(y-x\right)-\left(y-z\right)}{\left(y-z\right)\left(y-x\right)}=\frac{1}{y-z}-\frac{1}{y-x}\)
\(\frac{x-y}{\left(z-x\right)\left(z-y\right)}=\frac{\left(z-y\right)-\left(z-x\right)}{\left(z-x\right)\left(z-y\right)}=\frac{1}{z-x}-\frac{1}{z-y}\)
Suy ra: \(\frac{y-z}{\left(x-y\right)\left(x-z\right)}+\frac{z-x}{\left(y-z\right)\left(y-x\right)}+\frac{x-y}{\left(z-x\right)\left(z-y\right)}\)
\(=\frac{1}{x-y}-\frac{1}{x-z}+\frac{1}{y-z}-\frac{1}{y-x}+\frac{1}{z-x}-\frac{1}{z-y}\)
\(=\frac{2}{x-y}+\frac{2}{y-z}+\frac{2}{z-x}\)
rồi bí mẹ chỗ này
P=\(\frac{x}{\left(x-y\right)\left(x-z\right)}+\frac{y}{\left(y-x\right)\left(y-z\right)}+\frac{z}{\left(z-y\right)\left(z-x\right)}\) =\(\frac{x}{\left(x-y\right)\left(x-z\right)}-\frac{y}{\left(x-y\right)\left(y-z\right)}+\frac{z}{\left(y-z\right)\left(x-z\right)}\) =\(\frac{x\left(y-z\right)-y\left(x-z\right)+z\left(x-y\right)}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}\) =\(\frac{xy-xz-xy+yz+xz-yz}{\left(x-y\right)\left(y-z\right)\left(x-z\right)}\) =0