X^2-Y^2-Z^2: Exploring the Equation

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Discussion Overview

The discussion revolves around the factorization of the expression x^2 - y^2 - z^2. Participants explore potential factorizations and the implications of different conditions on the terms involved.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant notes the standard factorization for x^2 - y^2 and questions how it extends to x^2 - y^2 - z^2.
  • Another participant suggests that the factorization may not yield the same results as with two terms, particularly if x^2 - y^2 is negative, leading to complications in standard factorization.
  • A different participant proposes a specific factorization of x^2 - y^2 - z^2 as (x+y+z)(x-y-z) + yz.
  • Another participant questions whether the proposed factorization should include a term of +2yz instead of +yz.

Areas of Agreement / Disagreement

Participants express differing views on the factorization of the expression, with no consensus reached on the correct form or the implications of the conditions on the terms.

Contextual Notes

Participants highlight the dependence of factorization on the signs and values of the terms involved, indicating that the standard approach may not apply universally.

JDude13
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So
[tex]x^2-y^2=(x+y)(x-y)[/tex]
in the same sense what does
[tex]x^2-y^2-z^2=?[/tex]
come to?
 
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JDude13 said:
So
[tex]x^2-y^2=(x+y)(x-y)[/tex]
in the same sense what does
[tex]x^2-y^2-z^2=?[/tex]
come to?

You are not going to necessarily get the kind of factorization you got with two terms. If your x^2 - y^2 was itself a positive number, you could apply the same formula that you used for x and y.

If however your x^2 - y^2 was negative you would get a negative term - a negative term which would be in the form -(a + b^2) (a, b^2 >= 0) which has no standard factorization.
 


Ive had a muck around with it and
[tex]x^2-y^2-z^2=(x+y+z)(x-y-z)+yz[/tex]
 


shouldn't that be +2yz?
 

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