  By using the complete discrimination system for polynomials, we study the
number of positive solutions in {\small $C[0,1]$} to the integral equation
{\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small
$k(x,y)=\phi_1(x)\phi_1(y)+\phi_2(x)\phi_2(y), \phi_i(x)>0, \phi_i(y)>0,
0<x,y<1, i=1,2,$} are continuous functions on {\small $[0,1]$}, {\small $n$} is
a positive integer. We prove the following results: when {\small $n= 1$},
either there does not exist, or there exist infinitely many positive solutions
in {\small $C[0,1]$}; when {\small $n\geq 2$}, there exist at least {\small 1},
at most {\small $n+1$} positive solutions in {\small $C[0,1]$}. Necessary and
sufficient conditions are derived for the cases: 1) {\small $n= 1$}, there
exist positive solutions; 2) {\small $n\geq 2$}, there exist exactly {\small
$m(m\in \{1,2,...,n+1\})$} positive solutions. Our results generalize the
existing results in the literature, and their usefulness is shown by examples
presented in this paper.
