If ` x + (1)/( x) = - 2 ` then what is the value of ` 1 + x^(3) + c^(6)` ?

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To solve the equation \( x + \frac{1}{x} = -2 \) and find the value of \( 1 + x^3 + x^6 \), we can follow these steps:

### Step 1: Solve for \( x \)
Given the equation:
\[ x + \frac{1}{x} = -2 \]
Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = -2x \]
Rearranging gives us:
\[ x^2 + 2x + 1 = 0 \]
This can be factored as:
\[ (x + 1)^2 = 0 \]
Thus, we find:
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Step 2: Calculate \( x^3 \) and \( x^6 \)
Now that we have \( x = -1 \), we can calculate \( x^3 \) and \( x^6 \):
\[ x^3 = (-1)^3 = -1 \]
\[ x^6 = (-1)^6 = 1 \]

### Step 3: Substitute into the expression
Now substitute \( x^3 \) and \( x^6 \) into the expression \( 1 + x^3 + x^6 \):
\[ 1 + x^3 + x^6 = 1 + (-1) + 1 \]
Simplifying this gives:
\[ 1 - 1 + 1 = 1 \]

### Final Answer
Thus, the value of \( 1 + x^3 + x^6 \) is:
\[ \boxed{1} \]