If $$ x^n+\dfrac{1}{x^{2n}}=1 $$ , find the value of the polynomial $$ x^{5n}+x^n+2 $$

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If $$ x^n+\dfrac{1}{x^{2n}}=1 $$  , find the value of the polynomial $$ x^{5n}+x^n+2 $$

Uniteasy Admin · Accepted Answer

$$ \because x^n+\dfrac{1}{x^{2n}}=1 $$
$$ 	herefore  x^{3n}=x^{2n}-1 $$
 $$ x^{5n}+x^n+2 $$
$$ = x^{3n}x^{2n}+x^n+2 $$
$$ =(x^{2n}-1)x^{2n}+x^n+2 $$
$$ =x^{4n}-x^{2n}+x^n+2 $$
$$ =x^{2n}(x^{2n}-1)+x^n+2 $$
$$ =x^{2n}(x^n-1)(x^n+1)+x^n+2 $$
$$ =-x^{2n} \dfrac{1}{x^{2n}}(x^n+1)+x^n+2 $$
$$ =-x^n-1+x^n+2 $$
$$ =1 $$

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