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\(x^2+xz-y^2-yz=\left(x^2-y^2\right)+xz-yz=\left(x-y\right)\left(x+y\right)+z\left(x-y\right)=\left(x-y\right)\left(x+y+z\right)\)
Tương tự: \(y^2+xy-z^2-xz=\left(y-z\right)\left(x+y+z\right)\)
\(z^2+yz-x^2-xy=\left(x+y+z\right)\left(z-x\right)\)
Khi đó:
\(P=\frac{1}{\left(y-z\right)\left(x-y\right)\left(x+y+z\right)}+\frac{1}{\left(z-x\right)\left(y-z\right)\left(x+y+z\right)}+\frac{1}{\left(x-y\right)\left(x+y+z\right)\left(z-x\right)}\)
\(=\frac{z-x+x-y+y-z}{\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(x+y+z\right)}=0\)
a) \(A=\frac{x\left(x^2-yz\right)}{x+y+z}+\frac{y\left(y^2-zx\right)}{x+y+z}+\frac{z\left(z^2-xy\right)}{x+y+z}\)
\(=\frac{x^3+y^3+z^3-3xyz}{x+y+z}\)
\(=\frac{\left(x+y+z\right)\left(x^2+y^2+z^2-xy-yz-xz\right)}{x+y+z}\)
\(=x^2+y^2+z^2-xy-yz-xz\)
b) \(B=\frac{2}{3}.\left[\frac{3}{4x^2+4x+4}+\frac{3}{4x^2-4x+4}\right]\)
\(=\frac{2}{3}.\frac{3}{4}.\left(\frac{1}{x^2+x+1}+\frac{1}{x^2-x+1}\right)\)
\(=\frac{1}{2}.\frac{x^2-x+1+x^2+x+1}{\left(x^2+x+1\right)\left(x^2-x+1\right)}\)
\(=\frac{1}{2}.\frac{2\left(x^2+1\right)}{\left(x^2+x+1\right)\left(x^2-x+1\right)}\)
\(=\frac{x^2+1}{\left(x^2+x+1\right)\left(x^2-x+1\right)}\)
(vì \(x^2+x+1=\left(x+\frac{1}{2}\right)^2+\frac{3}{4}>0\)
và \(x^2-x+1=\left(x-\frac{1}{2}\right)^2+\frac{3}{4}>0\))
