If `alpha&beta` are roots of the equations `3x^(2)+8x+2=0`, find the values of `alpha^(3)+beta^(3)`

Options

A. `245//18`
B. `-368//27`
C. `-219//27`
D. none of these

Answer

To find the value of \( \alpha^3 + \beta^3 \) given that \( \alpha \) and \( \beta \) are the roots of the equation \( 3x^2 + 8x + 2 = 0 \), we will use the identities related to the sum and product of roots.

### Step-by-Step Solution:

**Step 1: Identify the coefficients of the quadratic equation.**
The given quadratic equation is:
\[ 3x^2 + 8x + 2 = 0 \]
Here, \( a = 3 \), \( b = 8 \), and \( c = 2 \).

**Step 2: Calculate the sum of the roots \( \alpha + \beta \).**
Using Vieta's formulas, the sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{8}{3} \]

**Step 3: Calculate the product of the roots \( \alpha \beta \).**
The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} = \frac{2}{3} \]

**Step 4: Use the identity for \( \alpha^3 + \beta^3 \).**
We can use the identity:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \]
Substituting the values we found:
\[ \alpha^3 + \beta^3 = \left(-\frac{8}{3}\right)^3 - 3\left(\frac{2}{3}\right)\left(-\frac{8}{3}\right) \]

**Step 5: Calculate \( \left(-\frac{8}{3}\right)^3 \).**
\[ \left(-\frac{8}{3}\right)^3 = -\frac{512}{27} \]

**Step 6: Calculate \( 3\alpha\beta(\alpha + \beta) \).**
First, calculate \( 3\left(\frac{2}{3}\right)\left(-\frac{8}{3}\right) \):
\[ 3\left(\frac{2}{3}\right)\left(-\frac{8}{3}\right) = 2 \cdot -\frac{8}{3} = -\frac{16}{3} \]

**Step 7: Combine the results.**
Now substituting back into the equation:
\[ \alpha^3 + \beta^3 = -\frac{512}{27} - \left(-\frac{16}{3}\right) \]
To combine these, convert \(-\frac{16}{3}\) to have a common denominator of 27:
\[ -\frac{16}{3} = -\frac{16 \cdot 9}{27} = -\frac{144}{27} \]
Thus:
\[ \alpha^3 + \beta^3 = -\frac{512}{27} + \frac{144}{27} = -\frac{368}{27} \]

### Final Answer:
\[ \alpha^3 + \beta^3 = -\frac{368}{27} \]