If `alpha`, `beta`, `gamma` are the roots of the equation `9x^(3)-7x+6=0` then the equation `x^(3)+Ax^(2)+Bx+C=0` has roots `3alpha+2`, `3beta+2`, `3gamma+2`, where

Options

A. `A=6`
B. `B=-5`
C. `C=24`
D. `A+B+C=23`

Explanation

To solve the problem step by step, we will follow the transformation of roots method.

### Step 1: Identify the original polynomial and its roots
We start with the polynomial given in the question:
\[ 9x^3 - 7x + 6 = 0 \]
Let the roots of this polynomial be \( \alpha, \beta, \gamma \).

### Step 2: Define the new roots
We need to find a new polynomial whose roots are given by:
\[ 3\alpha + 2, \quad 3\beta + 2, \quad 3\gamma + 2 \]
Let's denote the new roots as \( p = 3\alpha + 2 \).

### Step 3: Express \( \alpha \) in terms of \( p \)
From the expression for \( p \), we can express \( \alpha \) as:
\[ \alpha = \frac{p - 2}{3} \]

### Step 4: Substitute \( \alpha \) into the original polynomial
Now, we substitute \( \alpha \) into the original polynomial:
\[ 9\left(\frac{p - 2}{3}\right)^3 - 7\left(\frac{p - 2}{3}\right) + 6 = 0 \]

### Step 5: Simplify the equation
Calculating \( \left(\frac{p - 2}{3}\right)^3 \):
\[ \left(\frac{p - 2}{3}\right)^3 = \frac{(p - 2)^3}{27} \]
Thus,
\[ 9\left(\frac{(p - 2)^3}{27}\right) = \frac{(p - 2)^3}{3} \]
Now, substituting back into the equation:
\[ \frac{(p - 2)^3}{3} - \frac{7(p - 2)}{3} + 6 = 0 \]
Multiplying through by 3 to eliminate the denominator:
\[ (p - 2)^3 - 7(p - 2) + 18 = 0 \]

### Step 6: Expand and simplify
Now, we expand \( (p - 2)^3 \):
\[ (p - 2)^3 = p^3 - 6p^2 + 12p - 8 \]
So the equation becomes:
\[ p^3 - 6p^2 + 12p - 8 - 7p + 18 = 0 \]
Combining like terms:
\[ p^3 - 6p^2 + 5p + 10 = 0 \]

### Step 7: Write the new polynomial
Now we have the polynomial:
\[ p^3 - 6p^2 + 5p + 10 = 0 \]
Replacing \( p \) with \( x \):
\[ x^3 - 6x^2 + 5x + 10 = 0 \]

### Step 8: Identify coefficients
From this polynomial, we can identify:
- \( A = -6 \)
- \( B = 5 \)
- \( C = 10 \)

### Step 9: Calculate \( A + B + C \)
Now, we calculate:
\[ A + B + C = -6 + 5 + 10 = 9 \]

### Final Result
Thus, the values are:
- \( A = -6 \)
- \( B = 5 \)
- \( C = 10 \)
- \( A + B + C = 9 \)