How can I compose these like (y^2-x^2)(z^2-x^2)(z^2-y^2)

[requied]

Homework Statement

This is expected from the question that compose this statement below.

Relevant Equations

factorizing

How can I transform

to

.
The question is not actually contain only this. But it's necessary to do homework like this .

 

[fresh_42]

Terms like $a^2-b^2$ can always be written as $a^2-b^2=(a+b)(a-b)$. Note that $y^4=(y^2)^2$ and now apply this to your expression.

 

[haruspex]

Homework Statement:: This is expected from the question that compose this statement below.
Relevant Equations:: factorizing

How can I transform View attachment 263425 to View attachment 263426.
The question is not actually contain only this. But it's necessary to do homework like this .

If you know that's what you need to turn it into, obviously you could work backwards, expanding the factorised form. So I assume you are asking how you could have found that factorisation for yourself.

The first thing to note is that all the powers in $x^2y^4-x^4y^2-x^2z^4+y^2z^4+x^4z^2-y^4z^2$ are even, so you can think of the variables as being $x^2, y^2, z^2$.
Next, because of the minus signs, I wouid check what happens if two of these are equal. With $x^2=y^2$ it is easily seen to collapse to zero, so you know $x^2-y^2$ is a factor. Etc.

If that had not worked, I would have collected up similar terms:
$x^2y^2(y^2-x^2)-x^2z^4+y^2z^4+x^4z^2-y^4z^2$
$x^2y^2(y^2-x^2)+(-x^2+y^2)z^4+(x^4-y^4)z^2$
Etc.