If one root of `x^3 - 12x^2 +kx -18=0` is thrice the sum of remaining two roots then k=

Options

A. a) `29`
B. b) `-29`
C. c) `19`
D. d) `15`

Answer

To solve the problem, we need to find the value of \( k \) in the polynomial equation \( x^3 - 12x^2 + kx - 18 = 0 \) given that one root is thrice the sum of the remaining two roots. Let's denote the roots of the equation as \( \alpha, \beta, \gamma \).

### Step 1: Use Vieta's Formulas
According to Vieta's formulas, the sum of the roots \( \alpha + \beta + \gamma \) is equal to the coefficient of \( x^2 \) with the opposite sign, which gives us:
\[ \alpha + \beta + \gamma = 12 \]

### Step 2: Establish the Relationship Between the Roots
We are given that one root (let's assume \( \alpha \)) is thrice the sum of the remaining two roots:
\[ \alpha = 3(\beta + \gamma) \]

### Step 3: Substitute \( \beta + \gamma \)
From the sum of the roots, we can express \( \beta + \gamma \) as:
\[ \beta + \gamma = 12 - \alpha \]
Substituting this into the equation from Step 2 gives us:
\[ \alpha = 3(12 - \alpha) \]

### Step 4: Solve for \( \alpha \)
Now we can solve for \( \alpha \):
\[ \alpha = 36 - 3\alpha \]
\[ \alpha + 3\alpha = 36 \]
\[ 4\alpha = 36 \]
\[ \alpha = 9 \]

### Step 5: Find \( \beta + \gamma \)
Now that we have \( \alpha \), we can find \( \beta + \gamma \):
\[ \beta + \gamma = 12 - \alpha = 12 - 9 = 3 \]

### Step 6: Use the Product of Roots
According to Vieta's formulas, the product of the roots \( \alpha \beta \gamma \) is equal to \(-(-18) = 18\):
\[ \alpha \cdot \beta \cdot \gamma = 18 \]
Substituting \( \alpha = 9 \):
\[ 9 \cdot \beta \cdot \gamma = 18 \]
\[ \beta \cdot \gamma = \frac{18}{9} = 2 \]

### Step 7: Form a Quadratic Equation
Now we have \( \beta + \gamma = 3 \) and \( \beta \cdot \gamma = 2 \). We can form a quadratic equation with these roots:
\[ x^2 - (\beta + \gamma)x + \beta \cdot \gamma = 0 \]
Substituting the values:
\[ x^2 - 3x + 2 = 0 \]

### Step 8: Solve the Quadratic Equation
We can factor this equation:
\[ (x - 1)(x - 2) = 0 \]
Thus, the roots are:
\[ \beta = 1, \quad \gamma = 2 \quad \text{(or vice versa)} \]

### Step 9: Substitute Back to Find \( k \)
Now we can substitute \( \alpha, \beta, \gamma \) back into the original polynomial to find \( k \):
\[ x^3 - 12x^2 + kx - 18 = 0 \]
Substituting \( x = 9 \):
\[ 9^3 - 12(9^2) + 9k - 18 = 0 \]
Calculating:
\[ 729 - 972 + 9k - 18 = 0 \]
\[ 9k - 261 = 0 \]
\[ 9k = 261 \]
\[ k = \frac{261}{9} = 29 \]

### Final Answer
Thus, the value of \( k \) is:
\[ \boxed{29} \]