If the digits of a number may be any one of 0 through 9 (except the first digit cannot be zero), and the system of alternating signs of coefficients is used to create a polynomial, what is the least number m of real zeros the polynomial can have? What is the greatest number M of real zeros it may have? Can a polynomial of this type have any whole number of real zeros between m and M? Why or why not?

To determine the least and greatest number of real zeros that a polynomial can have, we can use the concept of Descartes' Rule of Signs. This rule helps us count the number of positive and negative real roots a polynomial might have.

First, let's consider the case of the least number of real zeros, denoted as m. To minimize the number of real zeros, we need to alternate the signs of the coefficients. Since the first digit cannot be zero, we can use the coefficients -1 and 1 alternately. Let's create a polynomial using these coefficients:
f(x) = -1x^n + 1x^(n-1) - 1x^(n-2) + 1x^(n-3) - ... + (-1)^(n-1)x - (-1)^(n)

Now, by Descartes' Rule of Signs, the number of positive or zero real roots of a polynomial is either equal to the number of sign changes in its coefficients or less than that by an even number. In our polynomial, the coefficient signs alternate, so there are no sign changes. Therefore, the number of positive or zero real roots is either 0 or an even number.

Similarly, when we consider the negative real roots, there are two possibilities: either 0 or an even number of them. Therefore, the least number of real zeros, m, is 0 in this case.

Next, let's consider the case of the greatest number of real zeros, denoted as M. To maximize the number of real zeros, we need to have as many sign changes in the coefficients as possible. Intuitively, if we have n coefficients, then the maximum number of sign changes is n.

Thus, the greatest number of real zeros, M, is equal to the number of coefficients in our polynomial.

Now, to address the last part of your question, can a polynomial of this type have any whole number of real zeros between m and M? The answer is yes. Since the number of positive and negative real roots is either 0 or an even number, we can have any even number of real zeros for this type of polynomial between m and M inclusive.

In conclusion:
- The least number of real zeros, m, is 0.
- The greatest number of real zeros, M, is equal to the number of coefficients in the polynomial.
- Any whole number of real zeros between m and M, inclusively, is possible for this type of polynomial.