Assertion (A) : If ` alpha , beta, gamma ` are the roots of ` x^3 -x-1=0` then ` alpha^3 + beta^3 + gamma^3 =1`
Reason (R ): If a +b+c=0 then ` a^3 + b^3 +c^3 = 3abc`

Options

A. both A and R are true R is the correct explanation of A
B. both A and R are true and R is not correct explanation of A
C. A is true and R is false
D. A is false and R is true

Answer

To solve the problem, we need to analyze both the assertion and the reason provided.

### Step 1: Analyze the Assertion
The assertion states that if \( \alpha, \beta, \gamma \) are the roots of the polynomial \( x^3 - x - 1 = 0 \), then:
\[ \alpha^3 + \beta^3 + \gamma^3 = 1 \]

### Step 2: Find the Sum of the Roots
Using Vieta's formulas, we know that for a cubic equation of the form \( x^3 + px^2 + qx + r = 0 \):
- The sum of the roots \( \alpha + \beta + \gamma = -p \)

In our equation \( x^3 - x - 1 = 0 \), there is no \( x^2 \) term, so:
\[ \alpha + \beta + \gamma = 0 \]

### Step 3: Use the Identity for the Sum of Cubes
We can use the identity for the sum of cubes:
\[ \alpha^3 + \beta^3 + \gamma^3 - 3\alpha\beta\gamma = (\alpha + \beta + \gamma)(\alpha^2 + \beta^2 + \gamma^2 - \alpha\beta - \beta\gamma - \gamma\alpha) \]
Since \( \alpha + \beta + \gamma = 0 \), we can simplify this to:
\[ \alpha^3 + \beta^3 + \gamma^3 = 3\alpha\beta\gamma \]

### Step 4: Find the Product of the Roots
Again using Vieta's formulas, the product of the roots \( \alpha \beta \gamma \) is given by:
\[ \alpha \beta \gamma = -\frac{r}{1} = -(-1) = 1 \]

### Step 5: Substitute the Product into the Identity
Now substituting \( \alpha \beta \gamma = 1 \) into the identity we derived:
\[ \alpha^3 + \beta^3 + \gamma^3 = 3 \cdot 1 = 3 \]

### Step 6: Conclusion on the Assertion
Thus, we find that:
\[ \alpha^3 + \beta^3 + \gamma^3 = 3 \]
This contradicts the assertion that \( \alpha^3 + \beta^3 + \gamma^3 = 1 \). Therefore, the assertion is false.

### Step 7: Analyze the Reason
The reason states that if \( a + b + c = 0 \), then:
\[ a^3 + b^3 + c^3 = 3abc \]
This is indeed true, as we derived this from the identity above.

### Final Conclusion
- Assertion (A) is **false**.
- Reason (R) is **true**.