Hmm. 1 -4x + x*x has 2 roots, 2+-sqrt(3) which are about 0.267949192431123 and 3.73205080756888.

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.