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Quotient of two polynomials

Source: VMO 2019

April 15, 2019
factoring polynomialspolynomialalgebra

Problem Statement

Consider polynomial f(x)=x2−αx+1f(x)={{x}^{2}}-\alpha x+1 with α∈R.\alpha \in \mathbb{R}. a) For α=152\alpha =\frac{\sqrt{15}}{2}, let write f(x)f(x) as the quotient of two polynomials with nonnegative coefficients. b) Find all value of α\alpha such that f(x)f(x) can be written as the quotient of two polynomials with nonnegative coefficients.
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