The equation `a_(8)x^(8)+a_(7)x_(8)^(7)+a_(6)x^(6)+…+a_(0)=0` has all its positive and real (where `a_(8)=1, a_(7)=-4, a_(0)=1//2^(8)`), then.

Options

A. `a_(1)=(1)/(2^(8))`
B. `a_(1)=-(1)/(2^(4))`
C. `a_(2)=(7)/(2^(4))`
D. `a_(3)=(1)/(2^(2))`

Answer

To solve the equation \( a_8 x^8 + a_7 x^7 + a_6 x^6 + \ldots + a_0 = 0 \) with the given coefficients \( a_8 = 1 \), \( a_7 = -4 \), and \( a_0 = \frac{1}{2^8} \), we need to analyze the roots of the polynomial.

### Step-by-Step Solution:

1. **Identify the Roots**:
The polynomial is of degree 8, which means it has 8 roots. Let's denote the roots as \( \alpha_1, \alpha_2, \ldots, \alpha_8 \).

2. **Use Vieta's Formulas**:
According to Vieta's formulas, for a polynomial \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0 = 0 \):
- The sum of the roots \( \alpha_1 + \alpha_2 + \ldots + \alpha_8 = -\frac{a_{n-1}}{a_n} = -\frac{-4}{1} = 4 \).
- The product of the roots \( \alpha_1 \alpha_2 \ldots \alpha_8 = (-1)^n \frac{a_0}{a_n} = (-1)^8 \frac{\frac{1}{2^8}}{1} = \frac{1}{2^8} \).

3. **Calculate the Arithmetic Mean (AM)**:
The arithmetic mean of the roots can be calculated as:
\[ AM = \frac{\alpha_1 + \alpha_2 + \ldots + \alpha_8}{8} = \frac{4}{8} = \frac{1}{2}. \]

4. **Calculate the Geometric Mean (GM)**:
The geometric mean of the roots is given by:
\[ GM = \sqrt[8]{\alpha_1 \alpha_2 \ldots \alpha_8} = \sqrt[8]{\frac{1}{2^8}} = \frac{1}{2}. \]

5. **Apply AM-GM Inequality**:
Since the arithmetic mean is equal to the geometric mean, we can conclude that all roots must be equal. Therefore, we have:
\[ \alpha_1 = \alpha_2 = \ldots = \alpha_8 = \frac{1}{2}. \]

6. **Verification**:
To verify, we can check the values:
- The sum of the roots: \( 8 \times \frac{1}{2} = 4 \) (correct).
- The product of the roots: \( \left(\frac{1}{2}\right)^8 = \frac{1}{256} \) (correct).

### Conclusion:
All roots of the polynomial are equal to \( \frac{1}{2} \).