AMC 12 2001 Problem 19

The polynomial $P(x)=x^3+ax^2+bx+c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=P(x)$ is 2, what is $b$?

$\text{(A) }-11 \text{(B) }-10 \text{(C) }-9 \text{(D) }1 \text{(E) }5$