x^3 + y^3 + z^3 ________ ________ _______ (x-y)(x-z) (y-z)(y-x) (z-x)(z-y) Упростите...

Решите задачу:

$\frac {x^3}{(x-y)(x-z)}+\frac{y^3}{ (y-z)(y-x)} +\frac{z^3}{(z-x)(z-y)}=$

$\frac {x^3}{(x-y)(x-z)}-\frac{y^3}{ (y-z)(x-y)} +\frac{z^3}{(x-z)(y-z)}= \frac {x^3(y-z)}{(x-y)(x-z)(y-z)}-\frac{y^3(x-z)}{ (y-z)(x-y)(x-z)} +\frac{z^3(x-y)}{(x-z)(y-z)(x-y)}=$

$\frac {(x^3y-z^3y)-(y^3x-y^3z)+(z^3x-x^3z)}{(x-z)(y-z)(x-z)}= \frac {y(x^3-z^3)+y^3(x-z)+zx(z^2-x^2)}{(x-z)(y-z)(x-z)}=$

$\frac {y(x-z)(x^2+xz+z^2)-y^3(x-z)-zx(x-z)(x+z)}{(x-z)(y-z)(x-z)}=$

$\frac {(x-z)(x^2y+yxz+yz^2)-y^3(x-z)-(x-z)(x^2z+z^2x)}{(x-z)(y-z)(x-z)}= \frac {(x-z)(x^2y+yxz+yz^2-y^3-x^2z-z^2x)}{(x-z)(y-z)(x-z)}=$

$\frac {x^2y+yxz+yz^2-y^3-x^2z-z^2x}{(x-y)(y-z)}= \frac {(x^2y-y^3)-(x^2z-yxz)-(z^2x-z^2y)}{(x-y)(y-z)}=$

$\frac {y(x^2-y^2)-xz(x-y)-z^2(x-y)}{(x-y)(y-z)}= \frac {y(x-y)(x+y)-xz(x-y)-z^2(x-y)}{(x-y)(y-z)}=$

$\frac {(x-y)(xy+y^2)-xz(x-y)-z^2(x-y)}{(x-y)(y-z)}= \frac {(x-y)(xy+y^2-xz-z^2)}{(x-y)(y-z)}=$

$\frac {xy+y^2-xz-z^2}{y-z}=\frac {(xy-xz)+(y^2-z^2)}{y-z}= \frac {x(y-z)+(y-z)(y+z)}{y-z}=\frac {x(y-z)+(y-z)(y+z)}{y-z}= \frac {(y-z)(x+y+z)}{y-z}=x+y+z$