<s>[INST] Proving that $x^{16} > 5$ when given a polynomial of degree $15$. I am unable to prove the following

If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x = 7$ prove that $x^{16} > 15$.

 [/INST] Note that your polynomial is a geometric series with first term $-x$, common ratio $-x^2$ so that it can be written as $$\frac{x(x^{16} - 1)}{x^2 + 1} = 7$$
So $$x^{16} = \frac{7(x^2 + 1)}{x} + 1$$
And now, all you need to do is show that $\frac{x^2 + 1}{x}  = x + \frac{1}{x} > 2$, which is not hard. 
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