<s>[INST] Factor $(x+y)^7-(x^7+y^7)$ I encountered the following problem while preparing for upcoming math contests.  

Factor $(x+y)^7-(x^7+y^7)$.

I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions said it's 
$$7xy(x+y)(x^2+xy+y^2)(x^2+xy+y^2)$$
Can someone explain how this is possible?
 [/INST] Expand $(x+y)^7$, you would get:
\begin{align*}
(x+y)^7-(x^7+y^7)&=x^7+7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6+y^7-x^7-y^7
\\ &=7x^6y+21x^5y^2+35x^4y^3+35x^3y^4+21x^2y^5+7xy^6
\\ &=7xy(x^5+3x^4y+5x^3y^2+5x^2y^3+3xy^4+y^5)
\\ &=7xy(x+y)(x^4+2x^3y+3x^2y^2+2x^3+y^4)
\\ &=7xy(x+y)(x^2+xy+y^2)^2,\text{which is your given answer.}
\end{align*}
</s>