### Instruction: The polynomial $P(x) = 2x^3 + ax^2 + bx + c$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = P(x)$ is 8. What is $b$?### Response: A polynomial of degree 3 has 3 zeros, so the mean of the zeros is the sum of the zeros divided by 3.
By Vieta's formulas, the sum of the zeros is $-a/2$ and the product of the zeros is $c/2$.
Therefore, the mean of the zeros is $-a/6$ and we have the equation $-a/6 = c/2$.
The sum of the coefficients is $2 + a + b + c$ and this is also equal to the mean and the product of the zeros, so we have another equation $2 + a + b + c = -a/6 = c/2$.
The $y$-intercept is the value of $P(0)$, which is $c$.
Given that $c = 8$, we can substitute this into the previous equations and simplify.
From $-a/6 = c/2$, we get $-a/6 = 4$ or $a = -24$.
From $2 + a + b + c = c/2$, we get $2 - 24 + b + 8 = 4$ or $b = 18$.
Therefore, the value of $b$ is 18.