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Question Number 191621 by Noorzai last updated on 27/Apr/23

Answered by gatocomcirrose last updated on 27/Apr/23

$${\mathrm{x}}^2=\mathrm{x}+1\Rightarrow {\mathrm{x}}^8={({\mathrm{x}}^2+2\mathrm{x}+1)}^2={(3\mathrm{x}+2)}^2$$$$={9\mathrm{x}}^2+12\mathrm{x}+4=9\mathrm{x}+9+12\mathrm{x}+4=21\mathrm{x}+13$$$${\mathrm{x}}^7={\left({\mathrm{x}}^2\right)}^3\mathrm{x}={(\mathrm{x}+1)}^3\mathrm{x}={\mathrm{x}}^4+{3\mathrm{x}}^3+{3\mathrm{x}}^2+\mathrm{x}$$$$={(\mathrm{x}+1)}^2+3\mathrm{x}(\mathrm{x}+1)+3(\mathrm{x}+1)+\mathrm{x}$$$$={4\mathrm{x}}^2+9\mathrm{x}+4=4(\mathrm{x}+1)+9\mathrm{x}+4=13\mathrm{x}+8$$$$$$$${\mathrm{x}}^8+{2\mathrm{x}}^7-47\mathrm{x}=21\mathrm{x}+13+26\mathrm{x}+16-47\mathrm{x}$$$$=29$$

Commented by Noorzai last updated on 28/Apr/23

$$thanks$$

Answered by manxsol last updated on 28/Apr/23

$$x^2=x+1\Rightarrow x^3=x^2+x$$$$x^3=2x+1\Rightarrow x^5=2x^3+x^2$$$$x^5=4x+2+x+1=5x+3$$$$x^7=5(2x+1)+3(x+1)=13x+8$$$$x^8=13(x+1)+8x=21x+13$$$$x^8-2x^7-47x=$$$$21x+13+26x+16-47x=29$$

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