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

[mathdad]

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?

 

[Opalg]

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.

 

[mathdad]

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

Correct?

 

[Opalg]

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?

 

[mathdad]

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.