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

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The equation x^2 - y^2 can be factored as (x+y)(x-y), but extending this to x^2 - y^2 - z^2 presents challenges. Unlike the two-term case, the three-term scenario does not yield a straightforward factorization. If x^2 - y^2 is positive, similar methods can be applied, but a negative result complicates the factorization process. The discussion suggests a potential factorization of x^2 - y^2 - z^2 as (x+y+z)(x-y-z) + yz, raising the question of whether it should include +2yz instead. Overall, the complexities of factoring three-variable equations are highlighted.
JDude13
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So
x^2-y^2=(x+y)(x-y)
in the same sense what does
x^2-y^2-z^2=?
come to?
 
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JDude13 said:
So
x^2-y^2=(x+y)(x-y)
in the same sense what does
x^2-y^2-z^2=?
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
x^2-y^2-z^2=(x+y+z)(x-y-z)+yz
 


shouldn't that be +2yz?
 

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