We are given the polynomial

f(x) = x x 11x+ 11x+ 19x19x 5x + 9

(a) What is the degree of this polynomial?

How many zeros can it have, in general?

Without knowing, at this stage, the exact values of the zeros of this polynomial, do they

have to be real and/or distinct?

(b)

What is the behavior of the polynomial as

x + and x ? Explain your answer.

(c) Describe, the rational zero theorem. Taking into account the form of the polynomial, can we use the rational zero theorem to find if the polynomial has rational

zeros? If yes, what are the possible zeros?

(d) Describe Descartes' rule of signs. Using this rule, what is the possible number of positive zeros of the polynomial, and what is the possible number of negative zeros?