Power Functions and Polynomial Functions

Question

2

Find the degree, leading coefficients, and the maximum number of real zeros of the polynomial.

Degree =   

Leading Coefficient =   

Maximum number of real zeros = 

Answer 1

The degree is 8

The leading coefficient is 1

The maximum number of real zeroes is 2

The final answer may come as a surprise, but we use DesCartes' rule of signs.

The maximum number of positive roots is 1; this is easy; however, note that, because we miss an x^6 term and the sign on the x^7 and x^5 terms are the same, we do not count a possible negative root there. We know we get the positive roots by the number of sign flips. However, the way to get the number of negative roots is to substitute -x for x and count the number of sign flips, not count the number of signs staying the same. If you have coefficients of all powers, the results are the same, but when you miss powers, results are different.