Factoring Fractional Expression

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Discussion Overview

The discussion revolves around factoring the fractional expression (x^3/8) - (512/x^3). Participants explore the steps involved in factoring the expression and whether further simplification is possible.

Discussion Character

  • Mathematical reasoning

Main Points Raised

  • One participant presents the initial factoring steps, identifying the least common denominator and simplifying the expression to [(x^3 - 64)(x^3 + 64)]/8x^3.
  • Another participant confirms the factorization of the numerator, relating it to the difference of squares and cubes.
  • Some participants question whether the expression can be reduced further, indicating that there may be additional factoring opportunities in the numerator.
  • A later reply suggests that there are both a sum and difference of cubes present, implying further factoring is possible.
  • One participant indicates they will complete the problem later, suggesting ongoing exploration of the topic.

Areas of Agreement / Disagreement

Participants generally agree on the initial steps of factoring but express uncertainty about whether the expression can be factored further. Multiple views on the extent of simplification remain unresolved.

Contextual Notes

There are indications of missing assumptions regarding the completeness of the factoring process and the potential for further simplification, which have not been fully addressed.

mathdad
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Factor (x^3/8) - (512/x^3).

LCD = 8x^3

(x^6 - 8(512))/8x^3

(x^6 - 4096)/8x^3

[(x^3 - 64)(x^3 + 64)]/8x^3

In the numerator, the expression (x^3 - 64) is the difference of cubes, right?
 
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Yes, in fact:

$$x^6-4096=x^6-2^{12}=\left(x^3\right)^2-\left(2^6\right)^2=\left(x^3+2^6\right)\left(x^3-2^6\right)=\left(x^3+\left(2^2\right)^3\right)\left(x^3-\left(2^2\right)^3\right)$$
 
Is that the final answer? Can we reduce it further?
 
RTCNTC said:
Is that the final answer? Can we reduce it further?

It can be factored further, as there is both a sum and difference of cubes there. Bear in mind I only dealt with the numerator. :D
 
I will complete the problem when time allows.
 

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