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f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}

Source: Irmo 2014 p2 q9

September 15, 2018
polynomialalgebra

Problem Statement

Let nn be a positive integer and a1,...,ana_1,...,a_n be positive real numbers. Let g(x)g(x) denote the product (x+a1)⋅...⋅(x+an)(x + a_1)\cdot ... \cdot (x + a_n) . Let a0a_0 be a real number and let f(x)=(x−a0)g(x)=xn+1+b1xn+b2xn−1+...+bnx+bn+1f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1} . Prove that all the coeffcients b1,b2,...,bn+1b_1,b_2,..., b_{n+1} of the polynomial f(x)f(x) are negative if and only if a0>a1+a2+...+ana_0 > a_1 + a_2 +...+ a_n.
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