Question

Chứng minh rằng:

\(\left(x+y\right)\left(y+z\right)\left(z+x\right)+xyz=\left(x+y+z\right)\left(xy+yz+zx\right)\)

Answer 1

Ta có :

\(VT=\left(x+y\right)\left(y+z\right)\left(z+x\right)+xyz\)

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

\(=xyz+y^2z+xz^2+yz^2+x^2y+y^2x+x^2z+xyz+xyz\)

\(=\left(x^2y+xyz+x^2z\right)+\left(y^2x+y^2z+xyz\right)+\left(xyz+z^2y+z^2x\right)\)\(=x\left(xy+yz+zx\right)+y\left(xy+yz+zx\right)+z\left(xy+yz+zx\right)\)

\(=\left(x+y+z\right)\left(xy+yz+zx\right)=VP\)

\(\left(đpcm\right)\)

:D