factoring in algebra

Scarlett

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Jan 9, 2009
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I need help in factoring out this problem.

(X^2 + y^2)^2 - (y^2 - z^2)^2

I started with this, but I'm not sure if this beginning is correct.

[(x^2+ y^2) - (y^2 -z^2)] [(x^2 + y^2) + (y^2 - z^2)]

What do I do?
 
What about this next:

Remove the parenthesis

[ x^2 + y^2 - y^2 - z^2][ x^2 + y^2 + y^2 - z^2]

Then

[ x^2 -z^2] [ x^2+ 2y^2 - z^2]

I still feel lost - I'm just not sure!
 
Scarlett said:
What about this next:

Remove the parenthesis

[ x^2 + y^2 - y^2 - z^2][ x^2 + y^2 + y^2 - z^2]<<<One wrong sign here that leads to the wrong sign below.

Then

[ x^2 -z^2] [ x^2+ 2y^2 - z^2] <<< One wrong sign. Make the correction and I think you are done.

I still feel lost - I'm just not sure!
 


I saw Loren's original post in this discussion; it looked fine to me, so I'm not sure about the misread.

You're not doing the subtraction properly in the first factor. You subtracted a negative z^2 term, but wrote it as -z^2.

Here is a symbolic example of the relevant rule when subtracting terms within parentheses:

A - (B - C) = A - B + C

The C ends up being positive, not negative. Subtracting a negative C is the same as adding a positive C.

You can think of that first minus sign as a -1 multiplying the contents of the parentheses, like this:

A + (-1)(B - C)

Now use the distributive property to get rid of the parentheses:

A + (-1)(B) + (-1)(-C)

A - B + C

Same result either way.

In the second factor, the -z^2 is not being subtracted, so I'm not sure why you changed it to +z^2.

I'm going back to your first result:

[(x^2+ y^2) - (y^2 -z^2)] [(x^2 + y^2) + (y^2 - z^2)]

Remove the inner grouping symbols.

(x^2 + y^2 - [y^2 - z^2])(x^2 + y^2 + y^2 - z^2)

Subtracting -z^2 is the same as adding +z^2.

(x^2 + y^2 - y^2 + z^2)(x^2 + y^2 + y^2 - z^2)

Let us know if you still do not understand why it's +z^2 in the first factor and -z^2 in the second factor.

 
Yes, I do see, and now I continue ???

I take this step
(x^2 + y^2 - y^2 + z^2)( x^2 +y^2 + y^2 - z^2)

and it becomes this???
(x^2 +z^2) (x^2 +2y^2 - z^2)

Is it now finished? or do I continue with this

(x + z)^2 (x+2y -z)^2

or what?
 
Yes, I do see, and now I continue ???

I take this step
(x^2 + y^2 - y^2 + z^2)( x^2 +y^2 + y^2 - z^2)

and it becomes this???
(x^2 +z^2) (x^2 +2y^2 - z^2)

Is it now finished? or do I continue with this

(x + z)^2 (x+2y -z)^2

or what?
 
Scarlett said:
(x^2 + y^2 - y^2 + z^2) (x^2 + y^2 + y^2 - z^2)

and it becomes this??? <<<< YES

(x^2 + z^2) (x^2 + 2y^2 - z^2)

Is it now finished? <<<< YES

(x + z)^2 (x+2y -z)^2<<<< NO NO NO!

Exponents are not "factorable" like that. You may not separate exponents away from their bases when trying to factor.



(x^2 + z^2) is called a "sum of squares".

A sum of squares is not factorable in the Real number system. (You might learn how to factor it later, if you study the Complex number system with its "imaginary" parts.)

 
Scarlett said:
What about this next:

Remove the parenthesis

[ x^2 + y^2 - y^2 + z^2][ x^2 + y^2 + y^2 - z^2]

Then

[ x^2 + z^2][ x^2+ 2y^2 - z^2]........That's it....
I still feel lost - I'm just not sure!
 
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