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Question-208814




Question Number 208814 by Ismoiljon_008 last updated on 23/Jun/24
Commented by Ismoiljon_008 last updated on 23/Jun/24
Help me please
$${Help}\:{me}\:{please} \\ $$
Commented by Ismoiljon_008 last updated on 23/Jun/24
If x^4 +x^3 +x^2 +x+1=0, then x^(71) −x^(61) +x^(51) +x^(41) +1=?
$${If}\:{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0},\:{then}\:{x}^{\mathrm{71}} −{x}^{\mathrm{61}} +{x}^{\mathrm{51}} +{x}^{\mathrm{41}} +\mathrm{1}=? \\ $$
Answered by Rasheed.Sindhi last updated on 23/Jun/24
If x^4 +x^3 +x^2 +x+1=0, then x^(71) −x^(61) +x^(51) +x^(41) +1=?   x^4 +x^3 +x^2 +x+1=0   ⇒(x−1)(x^4 +x^3 +x^2 +x+1)=0  x^5 −1=0⇒x^5 =1   x^(71) −x^(61) +x^(51) +x^(41) +1  =(x^5 )^(14) ∙x−(x^5 )^(12) ∙x+(x^5 )^(10) ∙x+(x^5 )^8 ∙x+1  =(1)^(14) ∙x−(1)^(12) ∙x+(1)^(10) ∙x+(1)^8 ∙x+1  =x−x+x+x+1=2x+1
$${If}\:{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0},\:{then}\:{x}^{\mathrm{71}} −{x}^{\mathrm{61}} +{x}^{\mathrm{51}} +{x}^{\mathrm{41}} +\mathrm{1}=? \\ $$$$\:{x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}=\mathrm{0} \\ $$$$\:\Rightarrow\left({x}−\mathrm{1}\right)\left({x}^{\mathrm{4}} +{x}^{\mathrm{3}} +{x}^{\mathrm{2}} +{x}+\mathrm{1}\right)=\mathrm{0} \\ $$$${x}^{\mathrm{5}} −\mathrm{1}=\mathrm{0}\Rightarrow{x}^{\mathrm{5}} =\mathrm{1} \\ $$$$\:{x}^{\mathrm{71}} −{x}^{\mathrm{61}} +{x}^{\mathrm{51}} +{x}^{\mathrm{41}} +\mathrm{1} \\ $$$$=\left({x}^{\mathrm{5}} \right)^{\mathrm{14}} \centerdot{x}−\left({x}^{\mathrm{5}} \right)^{\mathrm{12}} \centerdot{x}+\left({x}^{\mathrm{5}} \right)^{\mathrm{10}} \centerdot{x}+\left({x}^{\mathrm{5}} \right)^{\mathrm{8}} \centerdot{x}+\mathrm{1} \\ $$$$=\left(\mathrm{1}\right)^{\mathrm{14}} \centerdot{x}−\left(\mathrm{1}\right)^{\mathrm{12}} \centerdot{x}+\left(\mathrm{1}\right)^{\mathrm{10}} \centerdot{x}+\left(\mathrm{1}\right)^{\mathrm{8}} \centerdot{x}+\mathrm{1} \\ $$$$={x}−{x}+{x}+{x}+\mathrm{1}=\mathrm{2}{x}+\mathrm{1} \\ $$
Commented by Ismoiljon_008 last updated on 23/Jun/24
Thank you very much
$${Thank}\:{you}\:{very}\:{much} \\ $$

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