If `alpha,beta,gamma` are the zeroes of the polynomial `f(x)=x^(3)-5x^(2)-2x+24` such that `alphabeta=12`, then

Options

A. `alpha+beta=7`
B. `alpha-beta=pm1`
C. `gamma=-2`
D. All of these

Answer

To solve the problem step by step, we will analyze the polynomial \( f(x) = x^3 - 5x^2 - 2x + 24 \) and the given conditions about its roots \( \alpha, \beta, \gamma \).

### Step 1: Identify the roots and their relationships
We know that \( \alpha, \beta, \gamma \) are the roots of the polynomial, and we are given that \( \alpha \beta = 12 \).

### Step 2: Use Vieta's formulas
According to Vieta's formulas for a cubic polynomial \( ax^3 + bx^2 + cx + d \):
- The sum of the roots \( \alpha + \beta + \gamma = -\frac{b}{a} \)
- The sum of the product of the roots taken two at a time \( \alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a} \)
- The product of the roots \( \alpha \beta \gamma = -\frac{d}{a} \)

For our polynomial:
- \( a = 1 \)
- \( b = -5 \)
- \( c = -2 \)
- \( d = 24 \)

### Step 3: Calculate \( \alpha + \beta + \gamma \)
Using Vieta's first formula:
\[ \alpha + \beta + \gamma = -\frac{-5}{1} = 5 \]

### Step 4: Calculate \( \alpha \beta \gamma \)
Using Vieta's third formula:
\[ \alpha \beta \gamma = -\frac{24}{1} = -24 \]
Since \( \alpha \beta = 12 \), we can find \( \gamma \):
\[ \gamma = \frac{-24}{\alpha \beta} = \frac{-24}{12} = -2 \]

### Step 5: Calculate \( \alpha + \beta \)
We already found \( \alpha + \beta + \gamma = 5 \). Substituting \( \gamma = -2 \):
\[ \alpha + \beta - 2 = 5 \implies \alpha + \beta = 5 + 2 = 7 \]

### Step 6: Calculate \( \alpha - \beta \)
We know:
- \( \alpha + \beta = 7 \)
- \( \alpha \beta = 12 \)

To find \( \alpha - \beta \), we can use the identity:
\[ (\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta \]
Substituting the known values:
\[ 7^2 = \alpha^2 + \beta^2 + 2 \cdot 12 \]
\[ 49 = \alpha^2 + \beta^2 + 24 \implies \alpha^2 + \beta^2 = 49 - 24 = 25 \]

Now using the identity:
\[ \alpha - \beta = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} \]
Substituting the values:
\[ \alpha - \beta = \sqrt{7^2 - 4 \cdot 12} = \sqrt{49 - 48} = \sqrt{1} = \pm 1 \]

### Conclusion
We have found:
1. \( \alpha + \beta = 7 \)
2. \( \alpha - \beta = \pm 1 \)
3. \( \gamma = -2 \)

Thus, all options given in the problem are correct.