<s>[INST] Divisibility of higher power polynomials I am stuck with this problem suggested to me by a friend . The problem reads,
The power of $x^2+xy+y^2$ by which the polynomial $(x+y)^7-x^7-y^7$ is divisible are ?
I tried it by writing $x^7+y^7$ expansion and simplifying the polynomials in factors but that doesn't help as the resulting expression is ugly . Someone kindly suggest a method to solve this problem.
 [/INST] By the Binomial theorem we obtain:
$$(x+y)^7-x^7-y^7=7xy(x^5+3x^4y+5x^3y^2+5x^2y^3+3xy^4+y^5)=$$
$$=7xy(x+y)(x^4-x^3y+x^2y^2-xy^3+y^4+3xy(x^2-xy+y^2)+5x^2y^2)=$$
$$=7xy(x+y)(x^4+2x^3y+3x^2y^2+2xy^3+y^4)=7xy(x+y)(x^2+xy+y^2)^2.$$
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