Question

Rút gọn C= [ (x^2-y^2)^3+(y^2-z^2)^3+(z^2-x^2)^3] / [ (x-y)^3 + (y-z)^3 + (z-x)^3 ]

Answer 1

Ta có: \(\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3\)

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

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

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

=-3(x-y)(x+y)(x-z)(x+z)(y-z)(y+z)

Ta có: \(\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3\)

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

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

=-3(x-y)(y-z)(x-z)

Ta có: \(C=\frac{\left(x^2-y^2\right)^3+\left(y^2-z^2\right)^3+\left(z^2-x^2\right)^3}{\left(x-y\right)^3+\left(y-z\right)^3+\left(z-x\right)^3}\)

\(=\frac{-3(x-y)(x+y)(x-z)(x+z)(y-z)(y+z)}{-3\left(x-y\right)\left(y-z\right)\left(x-z\right)}=\left(x+y\right)\left(x+z\right)\left(y+z\right)\)