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Polynomial equation involving x^{n + 1} and (x+1)^{n + 1}

Source: Tuymaada 2020 Senior, P8

October 6, 2020
algebrapolynomial

Problem Statement

The degrees of polynomials PP and QQ with real coefficients do not exceed nn. These polynomials satisfy the identity P(x)xn+1+Q(x)(x+1)n+1=1. P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. Determine all possible values of Q(−12)Q \left( - \frac{1}{2} \right).
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