so, it boils down to the definition of a polynomial.
a polynomial is just a term with coefficients and variables at whatever power, however, it cannot be a rational, namely a fraction.
if we choose any positive exponent or that "x", we'll....let's pick say 3 and 7 just so you see what happens.
[tex]\bf 3x^4+4x^2-9x^{\setlength{\fboxsep}{1pt} -\boxed{\text{\tiny 3}}}+2\implies 3x^4+4x^2-9\cdot \cfrac{1}{x^3}+2\impliedby \textit{a fraction}
\\\\\\
3x^4+4x^2-9x^{\setlength{\fboxsep}{1pt} -\boxed{\text{\tiny 7}}}+2\implies 3x^4+4x^2-9\cdot \cfrac{1}{x^7}+2\impliedby \textit{another fraction}[/tex]
so, if we give it any positive exponent, we'll end up with a rational, however if we give it a negative, like say -9,
[tex]\bf 3x^4+4x^2-9x^{\setlength{\fboxsep}{1pt} -\boxed{\text{\tiny -9}}}+2\implies 3x^4+4x^2-9\cdot x^{+9}+2
\\\\\\
3x^4+4x^2-9x^9+2\impliedby \textit{not a fraction, thus a polynomial}[/tex]
