Prove-that-x-5-3x-4-17x-3-x-2-3x-17-cannot-be-factorized-completely-over-the-set-of-polynomials-with-integral-coefficients-

Question Number 110420 by Aina Samuel Temidayo last updated on 28/Aug/20

Prove that

x^{5} - {3 x}^{4} - {17 x}^{3} - x^{2} - 3 x + 17 cannot be

factorized completely over the set of

polynomials with integral coefficients .

Answered by 1549442205PVT last updated on 29/Aug/20

For this, it is enough to prove that

Polynomial P (x) = x^{5} - {3 x}^{4} - {17 x}^{3} - x^{2} - 3 x + 17

has no any integer roots

Hence, it is enough to

prove that all divisors of 17 {aren}^{'} t

roots of P (x) . This is always true by

checking directly

Commented by Aina Samuel Temidayo last updated on 29/Aug/20

Thanks, but how do you know it has

no integer roots ?

Commented by 1549442205PVT last updated on 30/Aug/20

Because all divisors {aren}^{'} t roots of

P (x) = 0

Commented by Aina Samuel Temidayo last updated on 30/Aug/20

What divisors ?