This question is part of an assignment of an institute in which I don't study .
Question: If p and q are polynomials with deg(q)>deg(p)+1 prove that sum of the residues of p/q at all its poles is 0.
Attempt: Question is related to Residue theorem . But no information is given about roots of q . Also If we try to use residue theorem at $C_{\infty}$ then p can have a root z =0 which will create problem as it is a pole at $\infty$ . So , $\sum_{k=1}^nRes[p/q;z_k]=-Res[p/q,\infty] $ which will be 0 only if $Res[p/q,\infty]$ =0 .
So , kindly tell how should I approach this problem.
A singularity of $(p/q)(z)$ at $z=\infty$ is the same thing as a singularity of $(p/q)(1/z)$ at $z=0$. Since we're dealing with a rational function, the singularity is either removable (in which case the residue is zero) or a pole. Given the assumption about degrees, what could the order of the pole be? What would be the associated residue?