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Bounded modulus in real-root polynom [Taiwan 2014 TST1 P1]

Source:

July 18, 2014
algebrapolynomialinequalitiesabsolute value

Problem Statement

Let f(x)=xn+an−2xn−2+an−3xn−3+⋯+a1x+a0f(x) = x^n + a_{n-2} x^{n-2} + a_{n-3}x^{n-3} + \dots + a_1x + a_0 be a polynomial with real coefficients (n≥2)(n \ge 2). Suppose all roots of ff are real. Prove that the absolute value of each root is at most 2(1−n)nan−2\sqrt{\frac{2(1-n)}n a_{n-2}}.
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