Multiply the equation by product_{i=1}^{n_c} (1+f_Rg*(K_i-1)). This gives you a polynomial of degree n_c-1 in f_Rg which can be solved using the "roots" function.
Or if you are lazy, just use:
K = [3 5 -7 9 -10 2];
c = [0.4 0.6 1.2 -0.7 1.1 0.8];
syms f_Rg
fun = sum(c.*(K-1)./(1+f_Rg*(K-1)));
f_Rg_num = double(solve(fun==0,f_Rg))
But don't use Newton's method in this case because the equation has nc-1 solutions, and it were pure luck if Newton's method converged to the "correct" one.
