If `alpha, beta` are the roots of the equation `ax^2 + bx +c=0` then the value of `(1+alpha+alpha^2)(1+beta+beta^2)` is

Options

A. 0
B. positive
C. negative
D. none of these

Answer

To find the value of \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \) where \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we can follow these steps:

### Step 1: Use Vieta's Formulas
From Vieta's formulas, we know:
\[ \alpha + \beta = -\frac{b}{a} \quad \text{and} \quad \alpha \beta = \frac{c}{a} \]

### Step 2: Expand the Expression
We need to expand the expression \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \):
\[ (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) = 1 + \alpha + \alpha^2 + \beta + \beta^2 + \alpha\beta + \alpha^2\beta + \alpha\beta^2 + \alpha^2\beta^2 \]

### Step 3: Group Terms
We can group the terms as follows:
\[ = 1 + (\alpha + \beta) + (\alpha^2 + \beta^2) + \alpha\beta + \alpha\beta(\alpha + \beta) + \alpha^2\beta^2 \]

### Step 4: Substitute Known Values
Now, we can substitute the known values from Vieta's formulas:
- \( \alpha + \beta = -\frac{b}{a} \)
- \( \alpha \beta = \frac{c}{a} \)

To find \( \alpha^2 + \beta^2 \), we use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values:
\[ \alpha^2 + \beta^2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \]

### Step 5: Substitute Everything Back
Now substituting everything back into the expression:
\[ 1 + \left(-\frac{b}{a}\right) + \left(\frac{b^2}{a^2} - \frac{2c}{a}\right) + \frac{c}{a} + \frac{c}{a}\left(-\frac{b}{a}\right) + \left(\frac{c}{a}\right)^2 \]

### Step 6: Simplify the Expression
Combining all the terms:
\[ = 1 - \frac{b}{a} + \frac{b^2}{a^2} - \frac{2c}{a} + \frac{c}{a} - \frac{bc}{a^2} + \frac{c^2}{a^2} \]
\[ = 1 - \frac{b}{a} - \frac{c}{a} + \frac{b^2 - bc + c^2}{a^2} \]

### Step 7: Final Expression
The final expression becomes:
\[ = \frac{a^2 - ab - ac + b^2 - bc + c^2}{a^2} \]

### Conclusion
Thus, the value of \( (1 + \alpha + \alpha^2)(1 + \beta + \beta^2) \) is:
\[ \frac{a^2 - ab - ac + b^2 - bc + c^2}{a^2} \]