Question

   x^3      +       y^3         +       z^3

________      ________        _______ 

(x-y)(x-z)        (y-z)(y-x)          (z-x)(z-y)

 

Упростите вырвжение

Answer 1

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

________      ________        _______          = ----------------------------------

(x-y)(x-z)        (y-z)(y-x)          (z-x)(z-y)         (x-y)(x-z)(z-y)

Answer 2

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

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

$frac {x^3(y-z)-y^3(x-z)+z^3(x-y)}{(x-z)(y-z)(x-z)}= frac {x^3y-x^3z-y^3x+y^3z+z^3x-z^3y}{(x-z)(y-z)(x-z)}=<var></var>$

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

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

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

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

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

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

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