Polynomials are linear combinations of positive integer powers. For example,
[tex]x^{100}+30x^2-\sqrt 2\,x+21[/tex]
is a polynomial, but
[tex]-x^{-1}-4x^\pi+x^{3/2}[/tex]
is not.
(In case you're unfamiliar with the term, a linear combination of two quantities is the sum of scaled versions of those quantities, e.g. a general linear combination of [tex]a[/tex] and [tex]b[/tex] is [tex]k_1a+k_2b[/tex], where [tex]k_1,k_2[/tex] are any numbers.)
[tex]8x^2-18\sqrt x+5x-3[/tex] is not a polynomial; the offending term is [tex]-18\sqrt x[/tex], or [tex]-18x^{1/2}[/tex].
