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Find the polynomial P [Indian RMO 2012(b) Q2]

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December 2, 2012
algebrapolynomialVietainequalitiesbinomial theoremsum of roots

Problem Statement

Let P(x)=xn+an−1xn−1+⋯+a0P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0 be a polynomial of degree n≥3.n\geq 3. Knowing that an−1=−(n1)a_{n-1}=-\binom{n}{1} and an−2=(n2),a_{n-2}=\binom{n}{2}, and that all the roots of PP are real, find the remaining coefficients. Note that (nr)=n!(n−r)!r!.\binom{n}{r}=\frac{n!}{(n-r)!r!}.
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