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Grade n polynomials

Source: Brazil Undergrad MO - Galois-Noether 2018 #21

November 26, 2019
polynomialBrazilian Math Olympiadcollege contestsalgebra

Problem Statement

Consider p(x)=xn+an−1xn−1+...+a1x+1 p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 a polynomial of positive real coefficients, degree n≥2 n \geq 2 e with n n real roots. Which of the following statements is always true?
a) p(2)<2(2n−1+1) p (2) <2 (2 ^ {n-1} +1) (b) p(1)<3 p (1) <3 c) p(1)>2n p (1)> 2 ^ n d) p(3)<3(2n−1−2) p (3 ) <3 (2 ^ {n-1} -2)
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