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 a polynomial of this type can have, we need to understand some rules about the number of zeros based on the polynomial's degree.
1. The least number of real zeros (m) can be found by considering the sign changes in the coefficients of the polynomial. The polynomial will have at most m positive real zeros.
2. The greatest number of real zeros (M) can be found by considering the sign changes in the coefficients of the polynomial along with the possibility of having complex conjugate pairs as zeros. The polynomial will have at most M real zeros, including both positive and negative.
3. A polynomial of this type cannot have any whole number of real zeros between m and M. The number of real zeros is restricted to integers only.
To find m, we need to consider the sign changes in the coefficients. Starting from the leftmost coefficient, we observe the changes in sign between neighboring coefficients. A sign change occurs when the signs of two neighboring coefficients are different. Then, m is equal to the number of sign changes or one less than that.
Similarly, to find M, we need to consider sign changes and the possibility of complex conjugate pairs. If there are k sign changes, then M is given by M = m + 2k, where k represents the number of complex conjugate pairs as zeros. Since complex conjugate pairs always come in pairs, we multiply the number of pairs by 2 and add it to m to get M.
Remember that a complex conjugate pair as a zero of a polynomial always counts as two separate real zeros.
To summarize:
- Find m by counting the number of sign changes in the coefficients.
- Determine k, the number of complex conjugate pairs, if any.
- Calculate M using the formula M = m + 2k.
Therefore, you can find the least and greatest number of real zeros a polynomial of this type can have by following these procedures.