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

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The discussion focuses on the mathematical expression (a + b + c)^3 - a^3 - b^3 - c^3 and the method to factor it. Participants agree that expanding (a + b + c)^3 is the initial step, allowing for the cancellation of -a^3, -b^3, and -c^3. The approach involves rewriting the expression as $[(a+b+c)^3 - a^3] - [b^3 + c^3]$, applying the difference of cubes to the first bracket and the sum of cubes to the second. This method leads to the cancellation of several terms, simplifying the expression significantly.

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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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