Thread: (a + b + c)^3

1. (a + b + c)^3

Verify this identity: (a + b + c)^3 + (a - b - c)^3 - (a + c - b)^3 - (a + b - c)^3 = 24abc

2. This is just a matter of expanding each bracketed expression, and then simplifying everything. Tedious, more than anything. Here's the first bracketed term expanded.

$(a + b + c)^3$

$=(a+b+c)(a^2 + ab + ac + ba + b^2 + bc + ca + cb + c^2)$

$=(a+b+c)(a^2 + 2ab + 2ac + b^2 + 2bc + c^2)$

$=a^3+3a^2 b+3a^2c+3ab^2+6abc+3ac^2+b^3+3b^2c+3b c^2+c^3$

3. Hello pacman
Originally Posted by pacman
Verify this identity: (a + b + c)^3 + (a - b - c)^3 - (a + c - b)^3 - (a + b - c)^3 = 24abc
First, can I encourage you to use LaTeX: it makes your questions and working much more readable. In this instance, all you need do to change your original expression to LaTeX is to select it, and then press the $\Sigma$ button on the toolbar. Which is what I've done here:

$(a + b + c)^3 + (a - b - c)^3 - (a + c - b)^3 - (a + b - c)^3 = 24abc$

Then: I think we can reduce the tedium a bit by noting one or two well-known results first:

$p^3 + q^3 = (p+q)(p^2-pq+q^2)$ (1)

$p^2-q^2 = (p-q)(p+q)$ (2)

$(p+q)^2+(p-q)^2 = 2p^2+2q^2$ (3)

We begin by re-grouping a little:

$(a+b+c)^3+(a-b-c)^3-(a+c-b)^3 -(a+b-c)^3$ $=\Big((a+[b+c])^3+(a-[b+c])^3\Big)-\Big((a+[c-b])^3 +(a-[c-b])^3\Big)$

Then, using (1) on each expression in the big brackets:

$= 2a\Big((a+[b+c])^2-(a+[b+c])(a-[b+c]) +(a-[b+c])^2\Big)$ - $2a\Big((a+[c-b])^2-(a+[c-b])(a-[c-b]) +(a-[c-b])^2\Big)$

$=2a\Big(2a^2 +2(b+c)^2 -a^2 +(b+c)^2\Big)-2a\Big(2a^2+2(c-b)^2-a^2+(c-b)^2\Big)$, using (2) and (3)

$=2a\Big(a^2 +3(b+c)^2\Big)-2a\Big(a^2+3(c-b)^2\Big)$

$=2a\Big(3(b+c)^2-3(c-b)^2\Big)$

$=6a(b^2+2bc+c^2-c^2+2bc-b^2)$

$=24abc$