Answer:
One possible polynomial is given by: [tex]f(x) = -x(x-3)^2[/tex]
Step-by-step explanation:
Zeros of a function:
Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
Single root at 0, double root at 3:
This means that:
[tex]f(x) = a(x - 0)(x - 3)(x - 3) = ax(x-3)^2[/tex]
Decreases as x approaches infinity.
This means that the leading coefficient should be negative. I am going to use -1. So
[tex]f(x) = -x(x-3)^2[/tex]
