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 the greatest number of real zeros the polynomial can have, we need to understand the relationship between the number of sign changes in the coefficients and the number of real zeros.

The sign changes in the coefficients refer to the number of times the signs alternate when writing out the coefficients. For example, if the polynomial is written with alternating signs like: $a_1x^n - a_2x^{n-1} + a_3x^{n-2} - a_4x^{n-3} + \ldots$, then there are sign changes between each term.

Now, let's consider the alternatives for the first coefficient $a_1$. Since the first digit cannot be zero, we have 9 possible choices for $a_1$.

We know that the number of sign changes in the coefficients of a polynomial is an upper bound on the number of real zeros. This means that the greatest number M of real zeros is equal to the number of sign changes in the coefficients.

To find the least number m of real zeros, we need to consider the extreme case where all sign changes happen, which results in the highest possible number of real zeros. So in this case, the least number m is equal to M + 1.

Now, let's analyze the number of sign changes for different values of $a_1$:

1. If $a_1$ is positive (greater than zero), there can be at most 9 sign changes.
2. If $a_1$ is negative, there can be at most 10 sign changes.

Therefore, the greatest number M of real zeros is 9 if $a_1$ is positive and 10 if $a_1$ is negative.

Now, let's consider whether a polynomial of this type can have any whole number of real zeros between m and M.

Since we are limited to using the digits 0 through 9 for the coefficients, there can only be a maximum of 9 sign changes. This means that there cannot be a polynomial with exactly 10 or more sign changes.

Therefore, a polynomial of this type can have any whole number of real zeros between m and M, as long as m is less than or equal to M.