Does adding to degree n polynomial terms with negative exponents still have n roots?

A polynomial has n roots. You can add terms less than n and still have a polynomial with n roots. Does this continue if you take negative exponents?

Eg. $$ax^2 + bx^1 + cx^0 + dx^{-1}...$$

Edit: I guess this isn't a polynomial per se, but does it still have n roots?

The Laurent polynomal $P(x) = ax^n + bx^{n-1} + cx^{n-2} + ... + ux^{-m+2} + vx^{-m+1} + wx^{-m}$ can be equivalently expressed as the quotient $\frac{ax^{m+n} + bx^{m+n-1} + cx^{m+n-2} + ... + ux^2 + vx + w}{x^m}$, whose roots are the roots of its numerator, a polynomial of degree $n + m$. The maximum number of roots is thus $n + m$, the sum of the most positive and most negative exponents.

Not true. For ex, $P(x)=x$ has only one root but $$ Q(x)=P(x)-3+\frac 1{x} $$ has two roots ($Q=0$ is a quadratic equation).