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. `-1`
B. `-1/2`
C. `1`
D. `1/2`

Explanation

To solve the problem step by step, we will analyze the given polynomial equation and use Vieta's formulas to find the values of \( c \) and \( d \), and then find the roots of the equation \( 2cx + d = 0 \).

### Step 1: Identify the given polynomial and its coefficients
The polynomial is given as:
\[ x^4 - 12x^3 + cx^2 + dx + 81 = 0 \]
From this, we can identify:
- Coefficient of \( x^4 \) (a) = 1
- Coefficient of \( x^3 \) (b) = -12
- Coefficient of \( x^2 \) (c) = c
- Coefficient of \( x^1 \) (d) = d
- Constant term = 81

### Step 2: Use Vieta's Formulas
According to Vieta's formulas, for a polynomial of the form \( ax^n + bx^{n-1} + ... + k = 0 \):
1. The sum of the roots \( x_1 + x_2 + x_3 + x_4 = -\frac{b}{a} \)
2. The sum of the product of the roots taken two at a time \( x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 = \frac{c}{a} \)
3. The sum of the product of the roots taken three at a time \( x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4 = -\frac{d}{a} \)
4. The product of the roots \( x_1x_2x_3x_4 = \frac{81}{1} = 81 \)

### Step 3: Calculate the sum of the roots
From Vieta's formulas:
\[ x_1 + x_2 + x_3 + x_4 = -\frac{-12}{1} = 12 \]

### Step 4: Calculate the product of the roots
The product of the roots is given as:
\[ x_1 x_2 x_3 x_4 = 81 \]

### Step 5: Apply AM-GM Inequality
Since the roots are positive, we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{x_1 + x_2 + x_3 + x_4}{4} \geq \sqrt[4]{x_1 x_2 x_3 x_4} \]
Substituting the known values:
\[ \frac{12}{4} \geq \sqrt[4]{81} \]
Calculating the right side:
\[ 3 \geq 3 \]
This shows that equality holds, which implies that all roots are equal. Therefore, we can denote:
\[ x_1 = x_2 = x_3 = x_4 = x \]

### Step 6: Find the value of the roots
From \( x_1 + x_2 + x_3 + x_4 = 12 \):
\[ 4x = 12 \implies x = 3 \]
Thus, the roots are \( x_1 = x_2 = x_3 = x_4 = 3 \).

### Step 7: Calculate \( c \)
Using the formula for the sum of the product of the roots taken two at a time:
\[ c = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_3 + x_2x_4 + x_3x_4 \]
Substituting the values:
\[ c = 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 + 3 \cdot 3 = 6 \cdot 9 = 54 \]

### Step 8: Calculate \( d \)
Using the formula for the sum of the product of the roots taken three at a time:
\[ d = -(x_1x_2x_3 + x_1x_2x_4 + x_1x_3x_4 + x_2x_3x_4) \]
Substituting the values:
\[ d = -\left(3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3 + 3 \cdot 3 \cdot 3\right) = -4 \cdot 27 = -108 \]

### Step 9: Find the roots of the equation \( 2cx + d = 0 \)
Substituting \( c = 54 \) and \( d = -108 \):
\[ 2(54)x - 108 = 0 \implies 108x - 108 = 0 \implies 108x = 108 \implies x = 1 \]

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