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First of all you need to know what a polynomial needs to have to be called a polynomial. 
A polynomial need to have the power of it's variables  as whole numbers, i.e non-negative. We can write x- \frac{1}{x} as x + x^{-1} . So this is correct that this is an expression with one variable [only x].  But since it has a (-1) power over x, it is not a polynomial.
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