Hello.
This is another problem from my book that I've been unable to solve.
Problem 122. Factor:
(b) $\displaystyle x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2);$
Thanks in advance.
Hello, DIOGYK!
$\displaystyle 122.\text{ Factor: }\;x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2)$
$\displaystyle \begin{array}{cc} \text{Expand:} & xy^2 - xz^2 + yz^2 - x^2y + x^2z - y^2z \\ \text{Rearrange:} & x^2z - x^2y -xz^2 + xy^2 + yz^2 - y^2z \\ \text{Factor:} & x^2(z-y) -x(z^2-y^2) + yz(z-y) \\ \text{Factor:} & x^2(z-y) - x(z-y)(z+y) - yz(z-y) \\ \text{Factor:} & (z-y)\big[x^2 - x(z+y) + yz \big] \\ \text{Expand:} & (z-y)(x^2-xz - xy + yz) \\ \text{Factor:} & (z-y)\big[x(x-z) - y(x-z)\big] \\ \text{Factor:} & (z-y)(x-z)(x-y) \\ \text{Rearrange:} & (x-y)(y-z)(z-x) \end{array}$