Question 1209771
Let's break down this problem step-by-step:

**1. Define the Zeros:**

* Let the zeros of p(x) be r, s, and t.

**2. Relationships Between Zeros and Coefficients:**

* **Mean of Zeros:** (r + s + t) / 3 = -a / 3
* **Product of Zeros:** rst = -c
* **Sum of Coefficients:** p(1) = 1 + a + b + c

**3. Given Conditions:**

* The mean of the zeros, the product of the zeros, and the sum of the coefficients are all equal.
* The y-intercept of p(x) is 0. This means p(0) = 0.

**4. Use the Y-Intercept Condition:**

* p(0) = 0^3 + a(0)^2 + b(0) + c = 0
* c = 0

**5. Apply the Equal Condition:**

* Since c = 0, the product of the zeros is rst = -c = 0. This implies at least one of the zeros is 0. Let's say r = 0.
* Now, we have:
    * (r + s + t) / 3 = -a / 3
    * rst = 0
    * 1 + a + b + c = 1 + a + b
* Since all three are equal:
    * -a / 3 = 0
    * 1 + a + b = 0

**6. Solve for a and b:**

* From -a / 3 = 0, we get a = 0.
* Substitute a = 0 into 1 + a + b = 0:
    * 1 + 0 + b = 0
    * b = -1

**7. Check the Mean of Zeros:**

* Since a = 0, the mean of the zeros is -a / 3 = 0.
* Since r = 0, we have (0 + s + t) / 3 = 0, which means s + t = 0.
* Also, 1 + a + b = 1 + 0 - 1 = 0, which confirms the condition.

**Answer:**

The value of b is -1.