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A polynomial with real roots...

Source: Bulgarian NMO 2017, 3rd round, p5

April 20, 2017
algebrapolynomialpolynomial equation

Problem Statement

Let nn be a natural number and f(x)f(x) be a polynomial with real coefficients having nn different positive real roots. Is it possible the polynomial: x(x+1)(x+2)(x+4)f(x)+ax(x+1)(x+2)(x+4)f(x)+a to be presented as the kk-th power of a polynomial with real coefficients, for some natural k≥2k\geq 2 and real aa?
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