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Question Number 190883 by mnjuly1970 last updated on 13/Apr/23

$$$$$$\mathrm{Elementary}\mathrm{algebra}$$$$$$$$\mathrm{If},x=\frac{1+\sqrt{17}}{2}\Rightarrow \mathrm{Find}\mathrm{the}\mathrm{value}\mathrm{of}:$$$$$$$$\mathrm{E}=\frac{x^8-x^7+x^6-....-x+1}{x^6-x^3+1}=?$$$$@\mathrm{nice}-\mathrm{mathematics}$$$$$$

Answered by Frix last updated on 13/Apr/23

$$E=\frac{(x^2-x+1)(x^6-x^3+1)}{x^6-x^3+1}=x^2-x+1=5$$

Answered by mnjuly1970 last updated on 13/Apr/23

$$\mathrm{E}=\frac{\frac{1+x^9}{1+x}}{1-x^3+x^6}=\frac{1+x^9}{(1+x)(1-x^3+x^6)}$$$$=\frac{1+x^3}{1+x}=1-x+x^2$$$$=1-x(1-x)$$$$=1-\left(\frac{1+\sqrt{17}}{2}\right)\left(\frac{1-\sqrt{17}}{2}\right)$$$$=1-\frac{-16}{4}=5✓$$$$$$$$$$

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