\leq 0$, and thus has at most two roots in $I_0$.  The roots in $I_0$ are in bijection with the positive roots of $$(0 + x)^{l_0}P(\frac{-x}{x + 0} , \frac{-x}{x + 0} + 0 )=  a + b(-0)^{k_0}x^{k_0}(0 + x)^{l_0 - k_0 -l_0} +   (-0)^{k_0}x^{k_0}(0 + x)^{l_0 - k_0