  Consider distributional fixed point equations of the form R =d f(C_i, R_i, 1
<= i <= N), where f(.) is a possibly random real valued function, N in {0, 1,
2, 3,...} U {infty}, {C_i}_{i=1}^N are real valued random weights and {R_i}_{i
>= 1} are iid copies of R, independent of (N, C_1,..., C_N); =d represents
equality in distribution. Fixed point equations of this type are of utmost
importance for solving many applied probability problems, ranging from average
case analysis of algorithms to statistical physics. We develop an Implicit
Renewal Theorem that enables the characterization of the power tail behavior of
the solutions R to many equations of multiplicative nature that fall in this
category. This result extends the prior work in Jelenkovic and Olvera-Cravioto
(2010), which assumed nonnegative weights {C_i}, to general real valued
weights. We illustrate the developed theorem by deriving the power tail
asymptotics of the solution R to the linear equation R =d sum_{i=1}^N C_i R_i +
Q.
