I a, b and c are not all equal and `alpha and beta` be the roots of the equation `ax^(2) + bx + c = 0` , then value of `(1 + alpha + alpha^(2)) (1 + beta + beta^(2))` is

Options

A. 0
B. positive
C. negative
D. non-negative

Answer

To solve the problem, we need to find the value of \((1 + \alpha + \alpha^2)(1 + \beta + \beta^2)\) given that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(ax^2 + bx + c = 0\).

### Step 1: Use the properties of roots
From Vieta's formulas, we know:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

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

### Step 3: Group the terms
Now, we can group the terms:
\[ = 1 + (\alpha + \beta) + (\alpha^2 + \beta^2) + (\alpha\beta + \alpha^2\beta + \alpha\beta^2 + \alpha^2\beta^2) \]

### Step 4: Substitute the known values
Using the known values from Vieta's formulas:
- Substitute \(\alpha + \beta = -\frac{b}{a}\)
- To find \(\alpha^2 + \beta^2\), we can use the identity:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2}{a^2} - \frac{2c}{a} \]

### Step 5: Substitute the product of roots
Now, we can substitute \(\alpha\beta\) into the expression:
\[ \alpha\beta + \alpha^2\beta + \alpha\beta^2 + \alpha^2\beta^2 = \alpha\beta(1 + \alpha + \beta) = \frac{c}{a}\left(1 - \frac{b}{a}\right) = \frac{c}{a} - \frac{bc}{a^2} \]

### Step 6: Combine all parts
Now we combine all parts:
\[ = 1 - \frac{b}{a} + \left(\frac{b^2}{a^2} - \frac{2c}{a}\right) + \left(\frac{c}{a} - \frac{bc}{a^2}\right) \]
\[ = 1 - \frac{b}{a} + \frac{b^2}{a^2} - \frac{2c}{a} + \frac{c}{a} - \frac{bc}{a^2} \]
\[ = 1 - \frac{b}{a} - \frac{c}{a} + \frac{b^2 - bc}{a^2} \]

### Final Expression
Thus, the final expression for \((1 + \alpha + \alpha^2)(1 + \beta + \beta^2)\) is:
\[ = 1 - \frac{b + c}{a} + \frac{b(b - c)}{a^2} \]