If `alpha,beta gamma` are zeroes of cubic polynomial `x^(3)+5x-2`, then find the value of `alpha^(3)+beta^(3)+gamma^(3)`.

Options

A. `2`
B. `3`
C. `6`
D. None

Explanation

To find the value of \( \alpha^3 + \beta^3 + \gamma^3 \) where \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 + 5x - 2 \), we can follow these steps:

### Step 1: Identify the coefficients of the polynomial
The given polynomial is \( x^3 + 5x - 2 \). We can compare this with the general form of a cubic polynomial \( ax^3 + bx^2 + cx + d \):
- \( a = 1 \)
- \( b = 0 \) (since there is no \( x^2 \) term)
- \( c = 5 \)
- \( d = -2 \)

### Step 2: Calculate the sum of the roots
The sum of the roots \( \alpha + \beta + \gamma \) for a cubic polynomial is given by the formula:
\[ \alpha + \beta + \gamma = -\frac{b}{a} \]
Substituting the values:
\[ \alpha + \beta + \gamma = -\frac{0}{1} = 0 \]

### Step 3: Calculate the sum of the product of the roots taken two at a time
The sum of the product of the roots taken two at a time \( \alpha\beta + \beta\gamma + \gamma\alpha \) is given by:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a} \]
Substituting the values:
\[ \alpha\beta + \beta\gamma + \gamma\alpha = \frac{5}{1} = 5 \]

### Step 4: Calculate the product of the roots
The product of the roots \( \alpha\beta\gamma \) is given by:
\[ \alpha\beta\gamma = -\frac{d}{a} \]
Substituting the values:
\[ \alpha\beta\gamma = -\frac{-2}{1} = 2 \]

### Step 5: Use the formula for the sum of cubes
The formula for the sum of cubes of the roots is:
\[ \alpha^3 + \beta^3 + \gamma^3 = (\alpha + \beta + \gamma)^3 - 3(\alpha + \beta)(\beta + \gamma)(\gamma + \alpha) \]
Since we already found \( \alpha + \beta + \gamma = 0 \), we can simplify:
\[ \alpha^3 + \beta^3 + \gamma^3 = 0^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) \]
Substituting the known values:
\[ \alpha^3 + \beta^3 + \gamma^3 = 0 - 3(5)(0) = 0 \]

### Step 6: Calculate the final value
Thus, we have:
\[ \alpha^3 + \beta^3 + \gamma^3 = 0 + 3 \cdot 2 = 6 \]

### Final Answer
The value of \( \alpha^3 + \beta^3 + \gamma^3 \) is \( 6 \).

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