If `x + (1)/(x)=1`, find
`x^(82) + x^(68) + x^(55) + x^(35) +x^(30) + x^(27) + x^(21) + x^(3) + 2`=?

Answer

To solve the equation \( x + \frac{1}{x} = 1 \) and find the value of \( x^{82} + x^{68} + x^{55} + x^{35} + x^{30} + x^{27} + x^{21} + x^{3} + 2 \), we can follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x + \frac{1}{x} = 1 \]
Multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = x \]
Rearranging gives:
\[ x^2 - x + 1 = 0 \]
Now, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -1, c = 1 \):
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
Thus, we have:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 2: Find \( x^3 \)
Next, we need to find \( x^3 \) using the identity:
\[ x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x}) \]
Substituting \( x + \frac{1}{x} = 1 \):
\[ x^3 + \frac{1}{x^3} = 1^3 - 3 \cdot 1 = 1 - 3 = -2 \]
This means:
\[ x^3 + \frac{1}{x^3} = -2 \]

### Step 3: Express higher powers of \( x \)
Using the relation \( x^3 + \frac{1}{x^3} = -2 \), we can express higher powers of \( x \) in terms of \( x^3 \):
- \( x^6 = (x^3)^2 \)
- \( x^9 = x^3 \cdot x^6 \)
- \( x^{12} = (x^3)^4 \)
- Continuing this, we can express \( x^{n} \) for \( n = 82, 68, 55, 35, 30, 27, 21, 3 \) in terms of \( x^3 \).

### Step 4: Calculate each term
We can express each exponent as multiples of 3:
- \( x^{82} = (x^3)^{27} \cdot x \)
- \( x^{68} = (x^3)^{22} \cdot x^2 \)
- \( x^{55} = (x^3)^{18} \cdot x \)
- \( x^{35} = (x^3)^{11} \cdot x^2 \)
- \( x^{30} = (x^3)^{10} \)
- \( x^{27} = (x^3)^9 \)
- \( x^{21} = (x^3)^7 \)
- \( x^{3} = (x^3)^1 \)

### Step 5: Substitute and simplify
Using the fact that \( x^3 + \frac{1}{x^3} = -2 \), we can substitute:
- \( x^{3k} = -1 \) for odd \( k \)
- \( x^{3k} = 1 \) for even \( k \)

Calculating:
- \( x^{82} + x^{68} + x^{55} + x^{35} + x^{30} + x^{27} + x^{21} + x^{3} + 2 \)
- Substitute \( x^{3k} \):
\[ (-1) + 1 + (-1) + 1 + 1 + (-1) + (-1) + (-1) + 2 \]
This simplifies to:
\[ -1 + 2 = 1 \]

### Final Answer
Thus, the value of \( x^{82} + x^{68} + x^{55} + x^{35} + x^{30} + x^{27} + x^{21} + x^{3} + 2 \) is:
\[ \boxed{0} \]