<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] A more compact display : If $σ_n = x^n + y^n$ and $P = xy$, then from 7-th power binomial expansion we get $$σ_1^7 = σ_7 + 7P(σ_5 + 3σ_3P + 5σ_1P^2)$$ and since $σ_1 | σ_n$for odd $n$, viz., $σ_3 = σ_1(σ_2 - P)$;$σ_5 = σ_3σ_2 - σ_1P^2 = σ_1\left(σ_2^2 - P(σ_2 + P)\right),$ $$σ_1^7 = σ_7 + 7σ_1P\left[σ_2^2 \overbrace{-\ P(σ_2 + P)} + \ 3P(σ_2 - P) + 5P^2\right]$$ $$= σ_7 + 7σ_1P\left[(σ_2 + P)(σ_2 + 2P) \overbrace{-\ P(σ_2 + P)}\right]$$ $$= σ_7 + 7σ_1P(σ_2 + P)^2,$$i.e., $$σ_1^7 - σ_7 = 7σ_1P(σ_2 + P)^2,$$ or more explicitly, $$(x + y)^7 - (x^7 + y^7) = 7xy(x + y)(x^2 + xy + y^2)^2.$$
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