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Find-an-integer-x-that-satisfies-the-equation-x-5-101x-3-999x-2-100900-0-

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Question Number 8707 by Sopheak last updated on 22/Oct/16
Find an integer x that satisfies the equation x^5 −101x^3 −999x^2 +100900=0
$${Find}\:{an}\:{integer}\:{x}\:{that}\:{satisfies}\:{the}\:{equation}\: \\ $$$${x}^{\mathrm{5}} −\mathrm{101}{x}^{\mathrm{3}} −\mathrm{999}{x}^{\mathrm{2}} +\mathrm{100900}=\mathrm{0} \\ $$
Answered by Rasheed Soomro last updated on 23/Oct/16
The solution is factor of 100900. So on trying x=10 is one solution. That also means that x−10 is a factor of x^5 −101x^3 −999x^2 +100900 By synthetic division we can determine other factor: determinant (((10)),1,0,(−101),(−999),0,(+100900)),(,,(10),(+100),(−10),(−10090),(−100900)),(,1,(10),(−1),(−1009),(−10090),([ 0))) So the other factor is x^4 +10x^3 −x^2 −1009x−10090 The rest roots of the given equation are the roots of x^4 +10x^3 −x^2 −1009x−10090=0 Trying all the possible integer factors of 10090 we learn that there is no other integer solution. So x=10
$$\mathrm{The}\:\mathrm{solution}\:\mathrm{is}\:\mathrm{factor}\:\mathrm{of}\:\mathrm{100900}. \\ $$$$\mathrm{So}\:\mathrm{on}\:\mathrm{trying}\:\mathrm{x}=\mathrm{10}\:\mathrm{is}\:\mathrm{one}\:\mathrm{solution}. \\ $$$$\mathrm{That}\:\mathrm{also}\:\mathrm{means}\:\mathrm{that}\:\mathrm{x}−\mathrm{10}\:\mathrm{is}\:\mathrm{a}\:\mathrm{factor} \\ $$$$\mathrm{of}\:\:{x}^{\mathrm{5}} −\mathrm{101}{x}^{\mathrm{3}} −\mathrm{999}{x}^{\mathrm{2}} +\mathrm{100900} \\ $$$$\mathrm{By}\:\mathrm{synthetic}\:\mathrm{division}\:\mathrm{we}\:\mathrm{can}\:\mathrm{determine} \\ $$$$\mathrm{other}\:\mathrm{factor}: \\ $$$$\begin{vmatrix}{\left.\mathrm{10}\right)}&{\mathrm{1}}&{\mathrm{0}}&{−\mathrm{101}}&{−\mathrm{999}}&{\mathrm{0}}&{+\mathrm{100900}}\\{}&{}&{\mathrm{10}}&{+\mathrm{100}}&{−\mathrm{10}}&{−\mathrm{10090}}&{−\mathrm{100900}}\\{}&{\mathrm{1}}&{\mathrm{10}}&{−\mathrm{1}}&{−\mathrm{1009}}&{−\mathrm{10090}}&{\left[\:\:\:\:\:\:\mathrm{0}\right.}\end{vmatrix} \\ $$$$\mathrm{So}\:\mathrm{the}\:\mathrm{other}\:\mathrm{factor}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{x}^{\mathrm{4}} +\mathrm{10x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} −\mathrm{1009x}−\mathrm{10090} \\ $$$$\mathrm{The}\:\mathrm{rest}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{given}\:\mathrm{equation}\:\mathrm{are}\:\mathrm{the} \\ $$$$\mathrm{roots}\:\mathrm{of}\:\:\:\mathrm{x}^{\mathrm{4}} +\mathrm{10x}^{\mathrm{3}} −\mathrm{x}^{\mathrm{2}} −\mathrm{1009x}−\mathrm{10090}=\mathrm{0} \\ $$$$\mathrm{Trying}\:\mathrm{all}\:\mathrm{the}\:\mathrm{possible}\:\mathrm{integer}\:\mathrm{factors}\:\mathrm{of}\:\mathrm{10090} \\ $$$$\mathrm{we}\:\mathrm{learn}\:\mathrm{that}\:\mathrm{there}\:\mathrm{is}\:\mathrm{no}\:\mathrm{other}\:\mathrm{integer}\:\mathrm{solution}. \\ $$$$\mathrm{So}\:\mathrm{x}=\mathrm{10} \\ $$$$ \\ $$

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