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Question Number 71327 by TawaTawa last updated on 13/Oct/19

Answered by mind is power last updated on 13/Oct/19

$$\mathrm{ferma}\mathrm{lemma}$$$${\mathrm{x}}^5=\mathrm{x}\left(5\right)$$$${\mathrm{x}}^4=1\left(5\right)\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\mathrm{x}\ne 0$$$$\mathrm{x}=0\mathrm{solution}$$$$\mathrm{x}\ne 0\mathrm{a}\iff 2\mathrm{x}+{2\mathrm{x}}^2-\mathrm{x}-{2\mathrm{x}}^2-\mathrm{x}=0\left(5\right)$$$$\Rightarrow 0=0\left(5\right)$$$$\Rightarrow \mathrm{always}\mathrm{true}$$$$\mathrm{b})\mathrm{x}\mathrm{solution}\Rightarrow {\mathrm{x}}^4+\mathrm{x}\ne 0\left(7\right)$$$$\mathrm{all}\mathrm{think}\mathrm{below}\mathrm{is}\mathrm{modd}\left(7\right)$$$${\mathrm{x}}^4+\mathrm{x}=\mathrm{x}(\mathrm{x}+1)({\mathrm{x}}^2-\mathrm{x}+1)=\mathrm{x}(\mathrm{x}+1)({\mathrm{x}}^2-\mathrm{x}-6)$$$$=\mathrm{x}(\mathrm{x}+1)(\mathrm{x}-3)(\mathrm{x}+2)=\mathrm{x}(\mathrm{x}-6)(\mathrm{x}-3)(\mathrm{x}-5)$$$${\mathrm{x}}^4+\mathrm{x}=0\left(7\right)\mathrm{if}\mathrm{x}=0,3,5,6\left(7\right)$$$$\mathrm{just}\mathrm{to}\mathrm{test}1,2,4$$$$\mathrm{fir}2\mid {\mathrm{x}}^4+\mathrm{x}=18=4\left(7\right)\mathrm{not}\mathrm{solution}$$$$\mathrm{x}=4\mid {\mathrm{x}}^4+\mathrm{x}+2=2^8+4+2=1\left(7\right)$$$$\mathrm{x}=1\Rightarrow {\mathrm{x}}^4+\mathrm{x}+2=4\left(7\right)\mathrm{not}\mathrm{slution}$$$$\mathrm{S}=\varnothing$$$$$$$$$$$$$$$$$$$$$$

Commented by TawaTawa last updated on 14/Oct/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by mind is power last updated on 14/Oct/19

$${\mathrm{y}}^{\prime }\mathrm{re}\mathrm{welcom}$$

Answered by delloptiplex last updated on 14/Oct/19

Commented by mathmax by abdo last updated on 14/Oct/19

$$thisintegralwassolvedseetheplatform.$$

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