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I have a task to find two polynomials $u(x)$ and $v(x)$ such that $$x^m u(x) + (1-x)^n v(x) = 1$$, how can I do that?

I understand that we can consider $x = 0$ and get $$v(0) = 0$$, similarly for $x = 1$ we get $$u(1) = 1$$, and then we could consider the formal derivatives of these polynomials according to the same principle, but I don’t understand how to bring this idea to a solution.

Robert Israel
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Андрей С.
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    Are $m$ and $n$ given natural numbers? Look up Bézout's identity and extended GCD algorithm – Robert Israel Feb 20 '24 at 15:14
  • @RobertIsrael, yes, there are natural, I read about Bézout's identity and GCD algorithm for polynomials, could you please write a little more detail on how I could apply them here? – Андрей С. Feb 20 '24 at 15:29

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