Question

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

Collected in the board: Nike function

Steven Zheng posted 2 months ago


Answer

\because x^n+\dfrac{1}{x^{2n}}=1

\therefore 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

Steven Zheng posted 2 months ago

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