# Show that (a+b+c)³ = a³ + b³ + c³ + (a+b+c)(ab+ac+bc)

As stated in the title, I'm supposed to show that $(a+b+c)³ = a³ + b³ + c³ + (a+b+c)(ab+ac+bc)$.

My reasoning: $$(a + b + c)³ = [(a + b) + c]³ = (a + b)³ + 3(a + b)²c + 3(a + b)c² + c³$$

$$(a + b + c)³ = (a³ + 3a²b + 3ab² + b³) + 3(a² + 2ab + b²)c + 3(a + b)c²+ c³$$

$$(a + b + c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3ab² + 3b²c + 3ac² + 3bc² + 6abc$$

$$(a + b + c)³ = (a³ + b³ + c³) + (3a²b + 3a²c + 3abc) + (3ab² + 3b²c + 3abc) + (3ac² + 3bc² + 3abc) - 3abc$$

$$(a + b + c)³ = (a³ + b³ + c³) + 3a(ab + ac + bc) + 3b(ab + bc + ac) + 3c(ac + bc + ab) - 3abc$$

$$(a + b + c)³ = (a³ + b³ + c³) + 3(a + b + c)(ab + ac + bc) - 3abc$$

$$(a + b + c)³ = (a³ + b³ + c³) + 3[(a + b + c)(ab + ac + bc) - abc]$$ It doesn't look like I made careless mistakes, so I'm wondering if the statement asked is correct at all.

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The statement is obviously wrong: look at it when $a=b=c=1$. –  KCd Jan 28 '13 at 14:43
You've probably not mentioned some condition? –  hjpotter92 Jan 28 '13 at 14:45
I transcribed the exercise. Found a typo or another at other parts of the book, so that's just another one of them. –  Sawyier Jan 28 '13 at 14:48
If it helps the final statement in your derivation looks correct. –  kaine Jan 28 '13 at 15:12

In general, $$a^n+b^n+c^n = \sum_{i+2j+3k=n} \frac{n}{i+j+k}\binom {i+j+k}{i,j,k} s_1^i(-s_2)^js_3^k$$

where $s_1=a+b+c$, $s_2=ab+ac+bc$ and $s_3=abc$ are the elementary symmetric polynomials.

In the case that $n=3$, the triples possible are $(i,j,k)=(3,0,0),(1,1,0),$ and $(0,0,1)$ yielding the formula:

$$a^3+b^3+c^2 = s_1^3 - 3s_2s_1 + 3s_3$$

which is the result you got.

In general, any symmetric homogeneous polynomial $p(a,b,c)$ of degree $n$ can be written in the form:

$$p(a,b,c)=\sum_{i+2j+3k=n} a_{i,j,k} s_1^i s_2^j s_3^k$$

for some constants $a_{i,j,k}$.

I've often thought Fermat's Last Theorem was most interesting when stated as a question about these polynomials. One statement of Fermat can be written as:

If $p$ is an odd prime, then $a^n+b^n+c^n=0$ if and only if $a+b+c=0$ and $abc=0$.

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Take $a=b=c=1$, you will find that that the expression is not correct.
You have a missing factor of $3$ on the right. –  O.L. Sep 9 '13 at 23:11