If `alpha ,beta and gamma` are the roots of
`x^3-2x^2+3x-4 =0 `, then find
(i) `sumalpha^2beta^2 (ii)sumalpha beta(alpha+beta)`

Options

A. 6
B. `-6`
C. `7`
D. `-7`

Answer

To solve the problem, we need to find two expressions based on the roots of the polynomial \( x^3 - 2x^2 + 3x - 4 = 0 \). The roots are denoted as \( \alpha, \beta, \) and \( \gamma \).

### Step 1: Identify the coefficients
The given polynomial can be compared to the general form \( ax^3 + bx^2 + cx + d = 0 \). From this, we can identify:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 3 \)
- \( d = -4 \)

### Step 2: Use Vieta's formulas
From Vieta's formulas, we can derive the following relationships among the roots:
1. The sum of the roots:
\[ \alpha + \beta + \gamma = -\frac{b}{a} = -\frac{-2}{1} = 2 \]
2. The sum of the products of the roots taken two at a time:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} = \frac{3}{1} = 3 \]
3. The product of the roots:
\[ \alpha\beta\gamma = -\frac{d}{a} = -\frac{-4}{1} = 4 \]

### Step 3: Find \( \sum \alpha^2 \beta^2 \)
We need to find \( \sum \alpha^2 \beta^2 \), which can be expressed as:
\[ \sum \alpha^2 \beta^2 = \alpha^2 \beta^2 + \beta^2 \gamma^2 + \gamma^2 \alpha^2 \]
This can be rewritten using the identity:
\[ \sum \alpha^2 \beta^2 = (\alpha\beta + \beta\gamma + \gamma\alpha)^2 - 2\alpha\beta\gamma(\alpha + \beta + \gamma) \]
Substituting the values we found:
\[ \sum \alpha^2 \beta^2 = (3)^2 - 2(4)(2) \]
Calculating this gives:
\[ = 9 - 16 = -7 \]

### Step 4: Find \( \sum \alpha \beta (\alpha + \beta) \)
Next, we need to find \( \sum \alpha \beta (\alpha + \beta) \):
\[ \sum \alpha \beta (\alpha + \beta) = \alpha\beta(\alpha + \beta) + \beta\gamma(\beta + \gamma) + \gamma\alpha(\gamma + \alpha) \]
This can be rewritten as:
\[ = \alpha\beta(2 - \gamma) + \beta\gamma(2 - \alpha) + \gamma\alpha(2 - \beta) \]
Expanding this gives:
\[ = 2(\alpha\beta + \beta\gamma + \gamma\alpha) - (\alpha\beta\gamma + \beta\gamma\alpha + \gamma\alpha\beta) \]
Substituting the known values:
\[ = 2(3) - 3(4) \]
Calculating this gives:
\[ = 6 - 12 = -6 \]

### Final Answers
Thus, the final results are:
1. \( \sum \alpha^2 \beta^2 = -7 \)
2. \( \sum \alpha \beta (\alpha + \beta) = -6 \)