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?

Question

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?