Question

What is
$x(y-z)(y+z)+y(z-x)+z(x-y)(x+y)$ equal to ?

• A. $(x+y)(y+z)(z+x)$
• B. $(x-y)(x-z)(z-y)$
• C. $(x+y)(z-y)(x-z)$
• D. $(y-x)(z-y)(x-z)$

Answer 1

The correct option is B $(x-y)(x-z)(z-y)$

$x(y-z)(y+z)+y(z+x)(z+x)+z(x-y)(x+y)$

$\Rightarrow x{y}^2-x{z}^2+(y{z}^2-y{x}^2)+z{x}^2-z{y}^2$

$\Rightarrow (x-z)(y^2+xz)+y(z-x)(z+x)$

$\Rightarrow (x-z)[y^2+xz-yz-xy]$

$\Rightarrow (x-z)\left[y\right(y-z)-x(y-z\left)\right]$

$\Rightarrow (x-z)(y-z)(y-x)$

$\Rightarrow (x-y)(x-z)(z-y)$