<s>[INST] Prove using factor theorem. Using factor theorem, show that $a+b$,$b+c$ and $c+a$ are factors of 
$(a + b + c)^3$ - $(a^3 + b^3 + c^3)$
How do we go about solving this ?
Thanks in advance !
 [/INST] To show, for example that $a+b$ is a factor of $p(a,b,c) = (a+b+c)^3 -(a^3 + b^3 + c^3)$, we consider $p$ as a polynomial in $a$, with constants $b$ and $c$, let's write $$ p_{b,c}(a) = p(a,b,c)$$
Now $a - (-b)$ is a factor of $p_{b,c}$ iff $-b$ is a root of $p_{b,c}$. We have 
\begin{align*}
  p_{b,c}(-b) &= (-b + b + c)^3 -\bigl( (-b)^3 + b^3 + c^3 \bigr) \\
               &= c^3 - c^3\\
               &= 0
\end{align*}
So $$ a+b \mid p_{b,c}(a) $$
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