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

## 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 .

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fresh_42
Mentor
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.

• berkeman
haruspex
Homework Helper
Gold Member
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.