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A famous Cauchy theorem for polynomials

Source: Moldova 2007 IMO-BMO TST IV problem 2

March 25, 2007
algebrapolynomialcalculusderivativeinductionalgebra proposed

Problem Statement

If b1,b2,…,bnb_{1}, b_{2}, \ldots, b_{n} are non-negative reals not all zero, then prove that the polynomial xn−b1xn−1−b2xn−2−…−bn=0x^{n}-b_{1}x^{n-1}-b_{2}x^{n-2}-\ldots-b_{n}=0 has only one positive root pp, which is simple. Moreover prove that any root of the polynomial does not exceed pp in absolute value.
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