If the roots of the equaion `x^4-12 x^3+c x^2+dx+81=0` are positive then the value of c is The value of d is. Roots of the equation 2cx+d=0 is

Options

A. `-27`
B. `-54`
C. `-81`
D. `-108`

Explanation

To solve the problem step by step, we will analyze the polynomial equation given and derive the values of \( c \) and \( d \) based on the conditions provided.

### Step 1: Identify the roots and their properties
The equation given is:

\[ x^4 - 12x^3 + cx^2 + dx + 81 = 0 \]

Let the roots of the equation be \( x_1, x_2, x_3, x_4 \). According to Vieta's formulas, we have:

1. \( x_1 + x_2 + x_3 + x_4 = 12 \) (sum of roots)
2. \( x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 = c \) (sum of the product of roots taken two at a time)
3. \( x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 = -d \) (sum of the product of roots taken three at a time)
4. \( x_1x_2x_3x_4 = 81 \) (product of roots)

### Step 2: Use the AM-GM inequality
Since the roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:

\[ \frac{x_1 + x_2 + x_3 + x_4}{4} \geq \sqrt[4]{x_1x_2x_3x_4} \]

Substituting the known values:

\[ \frac{12}{4} \geq \sqrt[4]{81} \]

Calculating the right side:

\[ 3 \geq 3 \]

Since the equality holds, all roots must be equal. Therefore, we have:

\[ x_1 = x_2 = x_3 = x_4 = 3 \]

### Step 3: Calculate the value of \( c \)
Now we can substitute the values of the roots into the equation for \( c \):

\[ c = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 \]

Calculating:

\[ c = 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 \]
\[ c = 6 \cdot 9 = 54 \]

### Step 4: Calculate the value of \( d \)
Next, we calculate \( d \):

\[ -d = x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 \]

Calculating:

\[ -d = 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 \]
\[ -d = 4 \cdot 27 = 108 \]

Thus, we find:

\[ d = -108 \]

### Step 5: Find the roots of the equation \( 2cx + d = 0 \)
Now we substitute the values of \( c \) and \( d \) into the equation \( 2cx + d = 0 \):

\[ 2(54)x - 108 = 0 \]
\[ 108x - 108 = 0 \]
\[ 108x = 108 \]
\[ x = 1 \]

### Final Answers
- The value of \( c \) is \( 54 \).
- The value of \( d \) is \( -108 \).
- The root of the equation \( 2cx + d = 0 \) is \( x = 1 \).