The polynomial `P(x)=x^(3)+ax^(2)+bx+c` has the property that the mean of its roots, the product of its roots, and the sum of its coefficients are all equal. If the `y`-intercept of the graph of `y=P(x)` is `2`,
The value of `P(1)` is

Options

A. `0`
B. `-1`
C. `2`
D. `-2`

Explanation

To solve the problem, we need to analyze the polynomial \( P(x) = x^3 + ax^2 + bx + c \) using the given conditions.

### Step 1: Understand the properties of the polynomial
1. The mean of the roots of the polynomial is given by:
\[ \text{Mean} = \frac{\alpha + \beta + \gamma}{3} \]
where \( \alpha, \beta, \gamma \) are the roots of the polynomial.

2. From Vieta's formulas, we know:
- The sum of the roots \( \alpha + \beta + \gamma = -a \)
- The product of the roots \( \alpha \beta \gamma = -c \)

### Step 2: Set up the equations based on the given conditions
From the problem, we have three quantities that are equal:
- Mean of the roots
- Product of the roots
- Sum of the coefficients

Thus, we can write:
\[ \frac{-a}{3} = -c = 1 + a + b + c \]

### Step 3: Use the y-intercept information
The y-intercept of the polynomial \( P(x) \) is given as \( P(0) = c \). Since it is stated that the y-intercept is 2, we have:
\[ c = 2 \]

### Step 4: Substitute \( c \) into the equations
Now substituting \( c = 2 \) into our previous equations:
1. From the product of the roots:
\[ -c = -2 \implies c = 2 \]

2. From the mean of the roots:
\[ \frac{-a}{3} = -2 \implies -a = -6 \implies a = 6 \]

3. Substitute \( a \) and \( c \) into the sum of coefficients:
\[ 1 + a + b + c = 1 + 6 + b + 2 = 9 + b \]
Setting this equal to \(-2\) (from the product of the roots):
\[ 9 + b = -2 \implies b = -2 - 9 = -11 \]

### Step 5: Calculate \( P(1) \)
Now we have \( a = 6 \), \( b = -11 \), and \( c = 2 \). We can calculate \( P(1) \):
\[ P(1) = 1^3 + 6 \cdot 1^2 - 11 \cdot 1 + 2 \]
\[ P(1) = 1 + 6 - 11 + 2 = -2 \]

### Final Answer
Thus, the value of \( P(1) \) is:
\[ \boxed{-2} \]