If two roots of ` x^3 - 9x^2 + 14 x +24 =0` are in the ratio 3:2 then the roots are

Options

A. `6,4,-1`
B. `3,2,4`
C. `1/2,1/3.(49)/(6)`
D. `6,4,2`

Answer

To solve the problem, we need to find the roots of the polynomial equation \( x^3 - 9x^2 + 14x + 24 = 0 \) given that two of the roots are in the ratio \( 3:2 \).

### Step-by-Step Solution:

1. **Assume the Roots**:
Let the two roots in the ratio \( 3:2 \) be \( 3k \) and \( 2k \). Let the third root be \( \alpha \).

2. **Use Vieta's Formulas**:
According to Vieta's formulas, the sum of the roots of the polynomial \( ax^3 + bx^2 + cx + d = 0 \) is given by:
\[ 3k + 2k + \alpha = -\frac{b}{a} \]
Here, \( a = 1 \) and \( b = -9 \), so:
\[ 5k + \alpha = 9 \]

3. **Find One Root by Trial**:
We can try to find one root by substituting values into the polynomial. Let's test \( x = -1 \):
\[ (-1)^3 - 9(-1)^2 + 14(-1) + 24 = -1 - 9 - 14 + 24 = 0 \]
Hence, \( x = -1 \) is a root, so we can set \( \alpha = -1 \).

4. **Substitute \( \alpha \) into the Sum of Roots**:
Now substituting \( \alpha = -1 \) into the sum of roots equation:
\[ 5k - 1 = 9 \]
Solving for \( k \):
\[ 5k = 10 \implies k = 2 \]

5. **Calculate the Roots**:
Now we can find the actual roots:
- The first root: \( 3k = 3 \times 2 = 6 \)
- The second root: \( 2k = 2 \times 2 = 4 \)
- The third root: \( \alpha = -1 \)

6. **Final Roots**:
Therefore, the roots of the equation \( x^3 - 9x^2 + 14x + 24 = 0 \) are \( 6, 4, \) and \( -1 \).

### Conclusion:
The roots of the equation are \( 6, 4, \) and \( -1 \).

---