What happens when you expand and factor (a + b + c)^3 - a^3 - b^3 - c^3?

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Discussion Overview

The discussion revolves around the expression (a + b + c)^3 - a^3 - b^3 - c^3, focusing on the methods of expansion and factoring. Participants explore the implications of applying the difference and sum of cubes in this context.

Discussion Character

  • Mathematical reasoning, Homework-related, Exploratory

Main Points Raised

  • Some participants propose expanding (a + b + c)^3 as a starting point to simplify the expression.
  • Others suggest that after expansion, terms -a^3, -b^3, and -c^3 can be canceled out.
  • A participant mentions applying the difference of cubes to [(a + b + c)^3 - a^3] and the sum of cubes to [b^3 + c^3].
  • There is a question about the outcome of applying these factoring techniques, with an expectation that many terms will cancel.

Areas of Agreement / Disagreement

Participants generally agree on the approach of expanding and factoring the expression, but the specific outcomes and further steps remain unresolved.

Contextual Notes

Participants have not completed the problem, and there are assumptions about the cancellation of terms that have not been explicitly verified.

mathdad
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Factor the expression.

(a + b + c)^3 - a^3 - b^3 - c^3

I believe the best way to tackle this problem is to expand (a + b + c)^3. By doing so, I should be able to cancel out -a^3 - b^3 - c^3, right?
 
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RTCNTC said:
Factor the expression.

(a + b + c)^3 - a^3 - b^3 - c^3

I believe the best way to start problem is by expanding
(a + b + c)^3. By doing so, I should be able to cancel out
-a^3 - b^3 - c^3, right?
Start by writing it as $[(a+b+c)^3 - a^3] - [b^3 + c^3]$, and factor the contents of each of the square brackets as the difference (or sum) of two cubes.
 
The difference of cubes is applied to
[(a + b + c)^3 - a^3] and the sum of cubes to [b^3 + c^3].

Correct?
 
RTCNTC said:
The difference of cubes is applied to
[(a + b + c)^3 - a^3] and the sum of cubes to [b^3 + c^3].

Correct?
Yes! What happens when you do that?
 
Opalg said:
Yes! What happens when you do that?

I have not completed the problem. So, my guess is that after applying the sum and difference of cubes, lots of terms will get canceled in the process.
 

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