1. ## Factoring

Hello.
This is another problem from my book that I've been unable to solve.

Problem 122. Factor:
(b) $x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2);$

2. ## Re: Factoring

Well isn't it obvious? $x = 4^2$ because $y^2$ told $x^2$ to lend him a power from $z^2;$

3. ## Re: Factoring

Hello, DIOGYK!

$122.\text{ Factor: }\;x(y^2-z^2)+y(z^2-x^2)+z(x^2-y^2)$

$\begin{array}{cc} \text{Expand:} & xy^2 - xz^2 + yz^2 - x^2y + x^2z - y^2z \\ \text{Rearrange:} & x^2z - x^2y -xz^2 + xy^2 + yz^2 - y^2z \\ \text{Factor:} & x^2(z-y) -x(z^2-y^2) + yz(z-y) \\ \text{Factor:} & x^2(z-y) - x(z-y)(z+y) - yz(z-y) \\ \text{Factor:} & (z-y)\big[x^2 - x(z+y) + yz \big] \\ \text{Expand:} & (z-y)(x^2-xz - xy + yz) \\ \text{Factor:} & (z-y)\big[x(x-z) - y(x-z)\big] \\ \text{Factor:} & (z-y)(x-z)(x-y) \\ \text{Rearrange:} & (x-y)(y-z)(z-x) \end{array}$

4. ## Re: Factoring

Thank you Soroban for very clear explanation. I should have tried more myself, this wasn't hard.

That's the problem uperkurk, $y^2$ didn't tell me what it told $x^2$.