Đáp án:
Giải thích các bước giải:
Ta có:
`\frac{y-z}{(x-y)(x-z)}+\frac{z-x}{(y-z)(y-x)}+\frac{x-y}{(z-x)(z-y)}`
`=\frac{-(y-z)}{(x-y)(z-x)}+\frac{-(z-x)}{(y-z)(x-y)}+\frac{-(x-y)}{(z-x)(y-z)}`
`=-\frac{(y-z)^2+(z-x)^2+(x-y)^2}{(x-y)(y-z)(z-x)}`
`=-\frac{y^2-2yz+z^2+z^2-2xz+x^2+x^2-2xy+y^2}{(x-y)(y-z)(z-x)}`
`=-\frac{2x^2+2y^2+2z^2-2xy-2yz-2xz}{(x-y)(y-z)(z-x)}`
`=\frac{2xy+2yz+2xz-2x^2-2y^2-2z^2}{(x-y)(y-z)(z-x)}(1)`
Lại có:
`\frac{2}{x-y}+\frac{2}{y-z}+\frac{2}{z-x}`
`=\frac{2(y-z)(z-x)+2(x-y)(z-x)+2(x-y)(y-z)}{(x-y)(y-z)(z-x)}`
`=\frac{2(yz-xy-x^2+xz)+2(xz-x^2-yz+xy)+2(xy-xz-y^2+yz)}{(x-y)(y-z)(z-x)}`
`=\frac{2yz-2xy-2x^2+2xz+2xz-2x^2-2yz+2xy+2xy-2xz-2y^2+2yz}{(x-y)(y-z)(z-x)}`
`=\frac{2xy+2yz+2xz-2x^2-2y^2-2z^2}{(x-y)(y-z)(z-x)}(2)`
Từ `(1);(2)⇒\frac{y-z}{(x-y)(x-z)}+\frac{z-x}{(y-z)(y-x)}+\frac{x-y}{(z-x)(z-y)}=\frac{2}{x-y}+\frac{2}{y-z}+\frac{2}{z-x}` (đpcm)
