TITLE: $p(x)x^n+q(x)(1-x)^n=1$ for some $p(x), q(x)\in \mathbb{Z}[x]$, what explicitly $p(x), q(x)$ are?
QUESTION [6 upvotes]: Because $x^n$ and $(x-1)^n$ are relatively prime in $\mathbb{Z}[x]$, so 
$p(x)x^n+q(x)(1-x)^n=1$ for some $p(x), q(x)\in \mathbb{Z}[x]$. What are the explicit formulas of $p(x), q(x)$?

REPLY [8 votes]: Hint:
Start with the equality $$x+(1-x)=1$$ raise to a large enough power ( $2n-1$ will do it ), and use Newton binomial formula. Separate the LHS into a multiple of $x^n$ and a multiple of $(1-x)^n$.