If `alpha, beta, gamma` are the roots of the equaion `x^(3)+px^(2)+qx+r=0`, then `(1-alpha^(2)) (1-beta^(2)) (1-gamma^(2))` is equal to

Options

A. `(1+q)^(2)-(p+r)^(2)`
B. `(1+q)^(2)+(p+r)^(2)`
C. `(1-q)^(2)+ (p-r)^(2)`
D. none of these

Answer

To solve the problem, we need to find the value of \( (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) \) where \( \alpha, \beta, \gamma \) are the roots of the cubic equation \( x^3 + px^2 + qx + r = 0 \).

### Step-by-Step Solution:

1. **Expand the Expression**:
\[ (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) = 1 - (\alpha^2 + \beta^2 + \gamma^2) + (\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2) - \alpha^2\beta^2\gamma^2 \]

2. **Using Vieta's Formulas**:
From Vieta's formulas for the roots of the polynomial \( x^3 + px^2 + qx + r = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -p \)
- The sum of the products of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha = q \)
- The product of the roots \( \alpha\beta\gamma = -r \)

3. **Calculate \( \alpha^2 + \beta^2 + \gamma^2 \)**:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha) \]
Substituting the values from Vieta's:
\[ = (-p)^2 - 2q = p^2 - 2q \]

4. **Calculate \( \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 \)**:
\[ \alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \]
Substituting the values:
\[ = q^2 - 2(-r)(-p) = q^2 - 2rp \]

5. **Calculate \( \alpha^2\beta^2\gamma^2 \)**:
\[ \alpha^2\beta^2\gamma^2 = (\alpha\beta\gamma)^2 = (-r)^2 = r^2 \]

6. **Substituting Back into the Expanded Expression**:
Now substituting all the calculated values back into the expanded expression:
\[ (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) = 1 - (p^2 - 2q) + (q^2 - 2rp) - r^2 \]
Simplifying this gives:
\[ = 1 - p^2 + 2q + q^2 - 2rp - r^2 \]

7. **Final Result**:
Thus, the final expression is:
\[ (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) = 1 + q^2 - p^2 - r^2 + 2q - 2rp \]