5

Part of 2017 Bulgaria National Olympiad

Problems(1)

A polynomial with real roots...

Source: Bulgarian NMO 2017, 3rd round, p5

n

be a natural number and

f(x)

be a polynomial with real coefficients having

n

different positive real roots. Is it possible the polynomial:

x(x+1)(x+2)(x+4)f(x)+a

to be presented as the

k

-th power of a polynomial with real coefficients, for some natural

k\geq 2

a

algebrapolynomialpolynomial equation