Factorize $x(y^2 – z^2) + y(z^2 – x^2) + z(x^2 – y^2)$

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I know the answer is $-(y-z)(x-y)(x-z) = (x-y)(y-z)(z-x)$ via WolframAlpha.

However, I don't why that's the solution nor how to get it "by hand". (nor what specific area of math I can find the proof and/or algorithm).

Answer 1

On expanding:

$x(y^2 - z^2) + y(z^2 - x^2) + z(x^2 - y^2)=xy^2-xz^2+yz^2-yx^2+zx^2-zy^2 $

$=xy^2-yx^2+yz^2-xz^2+zx^2-zy^2$

$=xy(y-x)+z^2(y-x)+z(x^2-y^2)$

$=(x-y)(zx-z^2+zy-xy)$

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

$=(x-y)(y-z)(z-x)$