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

Options

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

Answer

To find the value of \( (1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2) \) where \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + px^2 + qx + r = 0 \), we can follow these steps:

### Step 1: Understand the expression
We need to evaluate the product \( (1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2) \). We can expand this expression using the distributive property.

### Step 2: Expand the expression
The expression can be expanded as follows:
\[ (1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2) = 1 + (\alpha^2 + \beta^2 + \gamma^2) + (\alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2) + \alpha^2 \beta^2 \gamma^2 \]

### Step 3: Use Vieta's Formulas
From Vieta's formulas, we know:
- \( \alpha + \beta + \gamma = -p \)
- \( \alpha \beta + \beta \gamma + \gamma \alpha = q \)
- \( \alpha \beta \gamma = -r \)

To find \( \alpha^2 + \beta^2 + \gamma^2 \), we can use the identity:
\[ \alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha \beta + \beta \gamma + \gamma \alpha) \]
Substituting the values from Vieta's:
\[ \alpha^2 + \beta^2 + \gamma^2 = (-p)^2 - 2q = p^2 - 2q \]

### Step 4: Calculate \( \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 \)
Using the identity:
\[ \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 = (\alpha \beta + \beta \gamma + \gamma \alpha)^2 - 2\alpha \beta \gamma(\alpha + \beta + \gamma) \]
Substituting the values from Vieta's:
\[ \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 = q^2 - 2(-r)(-p) = q^2 - 2pr \]

### Step 5: Calculate \( \alpha^2 \beta^2 \gamma^2 \)
This can be computed as:
\[ \alpha^2 \beta^2 \gamma^2 = (\alpha \beta \gamma)^2 = (-r)^2 = r^2 \]

### Step 6: Substitute back into the expanded expression
Now, we can substitute back into our expanded expression:
\[ (1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2) = 1 + (p^2 - 2q) + (q^2 - 2pr) + r^2 \]
Combining these terms gives:
\[ = 1 + p^2 - 2q + q^2 - 2pr + r^2 \]

### Final Result
Thus, the final value of \( (1 + \alpha^2)(1 + \beta^2)(1 + \gamma^2) \) is:
\[ 1 + p^2 - 2q + q^2 - 2pr + r^2 \]