But a pair of related statements are true.

The simpler to prove is that if the constant term is negative, and the others are nonnegative, then there is only one real root. Existence is easy to prove. Uniqueness falls out of the fact that the derivative is positive over the real numbers, so the function is monotone increasing. Therefore it can only be zero once.

The harder variation is that if the largest power has a negative coefficient and the others are non-negative then there is only one positive real root. Proving this directly gets messy. The short solution is to substitute y = 1/x and then multiply by a high enough power of y to get back to the former case with a negative constant coefficient and all of the others non-negative. This mapping is a bijection on the positive reals, and we are back to the easier statement to prove.

If you're trying to model something like an investment in a bond, this is sufficient to prove uniqueness. However with more complicated cash flows, you've got a complicated problem.

I have series of irregular negative payments followed by one positive payment. I would reorganize the data somewhat if I could solve the more general case where some of the intermediate payments were also positive, but this isn't necessary.

I hadn't suspected, until a few days ago, that this was a difficult problem. All the solutions I have looked at so far recognize that they will not always provide an answer, even when there is one. They generally say little or nothing about when there is or isn't a solution or the circumstances under which they will fail to find an existing answer. It would be helpful to know more about such constraints.

Unfortunately, I am not a mathematician and have difficulty understanding many of the texts I have found. What I need are some simple guidelines and constraints to go along with the algorithms.

To put this in perspective - the Excel XIRR function has been accepted as a solution thus far. But I don't see how to reliably find the maximum and minimum valued solutions where there are more than one and I don't know how likely it is to fail or what characteristics of the data would induce failure (other than leading payments of 0, which it doesn't handle at all).

As long as the one positive payment follows all of the negative payments, I guarantee that you only have one positive real root. See Re^2: Internal Rate of Return for proof. (I have the signs the other way there, but it is easy to switch them.)

Incidentally a leading payment of 0 is handled by dropping all of the payments of 0. Then either you have no payments, or you've got a leading payment that isn't 0.

I'll have to look up bijection, but I follow your explanation of the conversion and I can understand that if all but the first coefficient are positive then the derivative is always positive so there can be only one zero. That confirms my suspicion - Thanks!!