Find the value of `alpha beta^(2)+beta alpha^(2),` if `alpha` and `beta` are the zeroes of polynomial `3x^(2)+4x+2.`

Options

A. `3/7`
B. `1/9`
C. `-8/9`
D. `7/8`

Answer

To find the value of \( \alpha \beta^2 + \beta \alpha^2 \) where \( \alpha \) and \( \beta \) are the roots of the polynomial \( 3x^2 + 4x + 2 \), we can follow these steps:

### Step 1: Rewrite the expression
We can factor the expression \( \alpha \beta^2 + \beta \alpha^2 \):
\[ \alpha \beta^2 + \beta \alpha^2 = \alpha \beta (\beta + \alpha) \]

### Step 2: Find the sum and product of the roots
For a quadratic polynomial of the form \( ax^2 + bx + c \):
- The sum of the roots \( \alpha + \beta \) is given by \( -\frac{b}{a} \)
- The product of the roots \( \alpha \beta \) is given by \( \frac{c}{a} \)

In our case, \( a = 3 \), \( b = 4 \), and \( c = 2 \).

### Step 3: Calculate the sum of the roots
Using the formula for the sum of the roots:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{4}{3} \]

### Step 4: Calculate the product of the roots
Using the formula for the product of the roots:
\[ \alpha \beta = \frac{c}{a} = \frac{2}{3} \]

### Step 5: Substitute the values into the expression
Now we can substitute the values of \( \alpha + \beta \) and \( \alpha \beta \) into the expression we factored earlier:
\[ \alpha \beta (\beta + \alpha) = \alpha \beta (\alpha + \beta) = \left(\frac{2}{3}\right) \left(-\frac{4}{3}\right) \]

### Step 6: Calculate the final value
Now, we multiply:
\[ \alpha \beta (\alpha + \beta) = \frac{2}{3} \times -\frac{4}{3} = -\frac{8}{9} \]

Thus, the value of \( \alpha \beta^2 + \beta \alpha^2 \) is:
\[ \boxed{-\frac{8}{9}} \]