Why couldn't this be factored further?

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AI Thread Summary
The expression s^4 - 625 can be factored as (s^2 + 25)(s^2 - 25), where s^2 - 25 further factors into (s + 5)(s - 5). The discussion highlights that while the program allows this factorization, it does not permit further factoring of (s^2 + 25) because it represents a sum of squares, which cannot be factored over the real numbers. It is noted that (s^2 + 25) could be factored using complex numbers, but that is outside the scope of the current problem. Ultimately, the focus is on recognizing the limitations of factoring sums of squares in real number contexts.
Peter Tran
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Homework Statement



Factor the following difference of two squares. Assume that variables represent whole numbers.

s^4 - 625

Homework Equations



(A)^2 - (B)^2

The Attempt at a Solution



s^4-625
(s^2)^2 - 25^2
(s^2+25)(s^2-25)
(s^2+25)(s+5)(s+5) - Correct

The "program" wants me to stop there, but why? Couldn't I just go ahead and continue factoring the (s^2+25) also?
 
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You can't factor a2+b2. You were able to factor difference of two squares only, not sum of two squares.

p.s. I think you made a typo, it should be (s2+25)(s+5)(s-5)
 
Peter Tran said:

Homework Statement



Factor the following difference of two squares. Assume that variables represent whole numbers.

s^4 - 625

Homework Equations



(A)^2 - (B)^2

The Attempt at a Solution



s^4-625
(s^2)^2 - 25^2
(s^2+25)(s^2-25)
(s^2+25)(s+5)(s+5) - Correct

The "program" wants me to stop there, but why? Couldn't I just go ahead and continue factoring the (s^2+25) also?

No. There is no way to factor s2 + 25 (or any sum of two squares in the form a2 + b2) under the real numbers. However, if you are familiar with complex numbers, then s2 + 25 could be factored further, but I don't know if you need to do that.
 
Oh that's right, I completley forgot that I could only factor A2 - B2.
Thanks.
 

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