`(a alpha x+alpha^(2)y+1)^(100)` is a polynomial in x and y. If the sum of the coefficients vanishes for some real `alpha`, then possible values of a are

Options

A. `-2`
B. 2
C. 1
D. none of these

Answer

To solve the problem, we need to find the values of \( a \) such that the sum of the coefficients of the polynomial \( (a \alpha x + \alpha^2 y + 1)^{100} \) vanishes for some real \( \alpha \).

### Step-by-Step Solution:

1. **Understanding the Polynomial**:
The polynomial given is \( (a \alpha x + \alpha^2 y + 1)^{100} \). The sum of the coefficients of a polynomial can be found by substituting \( x = 1 \) and \( y = 1 \).

2. **Substituting Values**:
Substitute \( x = 1 \) and \( y = 1 \):
\[ (a \alpha \cdot 1 + \alpha^2 \cdot 1 + 1)^{100} = (a \alpha + \alpha^2 + 1)^{100} \]

3. **Setting the Sum of Coefficients to Zero**:
We want the sum of the coefficients to vanish, which means:
\[ a \alpha + \alpha^2 + 1 = 0 \]

4. **Rearranging the Equation**:
Rearranging gives us:
\[ a \alpha + \alpha^2 + 1 = 0 \implies \alpha^2 + a \alpha + 1 = 0 \]

5. **Finding Conditions for Real Roots**:
For this quadratic equation in \( \alpha \) to have real roots, the discriminant must be non-negative:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = a \), and \( c = 1 \). Thus, we have:
\[ D = a^2 - 4 \cdot 1 \cdot 1 = a^2 - 4 \]

6. **Setting the Discriminant to Zero**:
For the quadratic to have real roots, we set the discriminant \( D \) to be greater than or equal to zero:
\[ a^2 - 4 \geq 0 \]

7. **Solving the Inequality**:
This can be factored as:
\[ (a - 2)(a + 2) \geq 0 \]
The critical points are \( a = -2 \) and \( a = 2 \).

8. **Finding the Intervals**:
We analyze the sign of the product:
- For \( a < -2 \), both factors are negative, so the product is positive.
- For \( -2 < a < 2 \), one factor is negative and the other is positive, so the product is negative.
- For \( a > 2 \), both factors are positive, so the product is positive.

Thus, the solution to the inequality is:
\[ a \leq -2 \quad \text{or} \quad a \geq 2 \]

### Conclusion:
The possible values of \( a \) for which the sum of the coefficients vanishes are:
\[ a \in (-\infty, -2] \cup [2, \infty) \]