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

$$\mathrm{Prove}\mathrm{that}$$$${\mathrm{x}}^5-{3\mathrm{x}}^4-{17\mathrm{x}}^3-{\mathrm{x}}^2-3\mathrm{x}+17\mathrm{cannot}\mathrm{be}$$$$\mathrm{factorized}\mathrm{completely}\mathrm{over}\mathrm{the}\mathrm{set}\mathrm{of}$$$$\mathrm{polynomials}\mathrm{with}\mathrm{integral}\mathrm{coefficients}.$$

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

$$\mathrm{For}\mathrm{this},\mathrm{it}\mathrm{is}\mathrm{enough}\mathrm{to}\mathrm{prove}\mathrm{that}$$$$\mathrm{Polynomial}\mathrm{P}\left(\mathrm{x}\right)={\mathrm{x}}^5-{3\mathrm{x}}^4-{17\mathrm{x}}^3-{\mathrm{x}}^2-3\mathrm{x}+17$$$$\mathrm{has}\mathrm{no}\mathrm{any}\mathrm{integer}\mathrm{roots}$$$$\mathrm{Hence},\mathrm{it}\mathrm{is}\mathrm{enough}\mathrm{to}$$$$\mathrm{prove}\mathrm{that}\mathrm{all}\mathrm{divisors}\mathrm{of}17{\mathrm{aren}}^{\prime }\mathrm{t}$$$$\mathrm{roots}\mathrm{of}\mathrm{P}\left(\mathrm{x}\right).\mathrm{This}\mathrm{is}\mathrm{always}\mathrm{true}\mathrm{by}$$$$\mathrm{checking}\mathrm{directly}$$

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

$$\mathrm{Thanks},\mathrm{but}\mathrm{how}\mathrm{do}\mathrm{you}\mathrm{know}\mathrm{it}\mathrm{has}$$$$\mathrm{no}\mathrm{integer}\mathrm{roots}?$$

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

$$\mathrm{Because}\mathrm{all}\mathrm{divisors}{\mathrm{aren}}^{\prime }\mathrm{t}\mathrm{roots}\mathrm{of}$$$$\mathrm{P}\left(\mathrm{x}\right)=0$$

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

$$\mathrm{What}\mathrm{divisors}?$$

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