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

Options

A. `(1+q)^(2))-(p+r)^(2)`
B. `(1+q)^(2)+(p+r)^(2)`
C. `(1-q)^(2)+(p-r)^(2)`
D. `(1+r)^(2)-(p+q)^(2)`

Explanation

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

### Step-by-Step Solution:

1. **Express the polynomial in terms of its roots**:
The polynomial can be expressed as:
\[ x^3 + px^2 + qx + r = (x - \alpha)(x - \beta)(x - \gamma) \]

2. **Evaluate the polynomial at \(x = 1\)**:
Substitute \(x = 1\) into the polynomial:
\[ 1^3 + p(1^2) + q(1) + r = 1 + p + q + r \]
This gives us:
\[ 1 + p + q + r = (1 - \alpha)(1 - \beta)(1 - \gamma) \]

3. **Evaluate the polynomial at \(x = -1\)**:
Substitute \(x = -1\) into the polynomial:
\[ (-1)^3 + p(-1)^2 + q(-1) + r = -1 + p - q + r \]
This gives us:
\[ -1 + p - q + r = (-1 - \alpha)(-1 - \beta)(-1 - \gamma) \]
Simplifying the right side, we have:
\[ (-1 - \alpha)(-1 - \beta)(-1 - \gamma) = -(1 + \alpha)(1 + \beta)(1 + \gamma) \]

4. **Multiply the two results**:
Now, we have two equations:
\[ (1 + p + q + r)(-1 + p - q + r) = (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) \]
Using the difference of squares:
\[ (1 + q)^2 - (p + r)^2 \]

5. **Final Result**:
Therefore, we find that:
\[ (1 - \alpha^2)(1 - \beta^2)(1 - \gamma^2) = (1 + q)^2 - (p + r)^2 \]

Thus, the answer is:
\[ \boxed{(1 + q)^2 - (p + r)^2} \]