Post Closed as "too localized" by Pete L. Clark, Andres Caicedo, Bill Johnson, Andy Putman, Charles Siegel

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Luke's polynomial question

For every positive integer A there exists a unique polynomial Q_A(X) of degree A-1 satisfying the identity

(1-x/2)^A Q_A(x) + (x/2)^A Q_A(2-x) - 1 = 0

How to prove this ?

A simple proof plz im a noob.

Luke