If `x + (1)/(x)=1`, find
`x^(103) + x^(100) + x^(90) + x^(87) + x^(60) + x^(57) + x^(9) + x^(6) + x^(3) + 3=`

Answer

To solve the equation \( x + \frac{1}{x} = 1 \) and find the value of \( x^{103} + x^{100} + x^{90} + x^{87} + x^{60} + x^{57} + x^{9} + x^{6} + x^{3} + 3 \), we will follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x + \frac{1}{x} = 1 \]
Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 + 1 = x \]
Rearranging gives us:
\[ x^2 - x + 1 = 0 \]

### Step 2: Find the roots of the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -1, c = 1 \):
\[ x = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This simplifies to:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 3: Calculate \( x^3 \)
We can find \( x^3 \) using the identity \( x + \frac{1}{x} = 1 \):
Cubing both sides:
\[ \left( x + \frac{1}{x} \right)^3 = 1^3 \]
Expanding the left-hand side:
\[ x^3 + 3x \cdot \frac{1}{x} \cdot (x + \frac{1}{x}) = 1 \]
This simplifies to:
\[ x^3 + 3 = 1 \implies x^3 = -2 \]

### Step 4: Find higher powers of \( x \)
Using the result \( x^3 = -2 \):
- \( x^6 = (x^3)^2 = (-2)^2 = 4 \)
- \( x^9 = x^3 \cdot x^6 = -2 \cdot 4 = -8 \)
- \( x^{60} = (x^3)^{20} = (-2)^{20} = 1048576 \)
- \( x^{57} = x^{54} \cdot x^3 = (x^3)^{18} \cdot x^3 = 1048576 \cdot (-2) = -2097152 \)
- \( x^{87} = x^{84} \cdot x^3 = (x^3)^{28} \cdot x^3 = 268435456 \cdot (-2) = -536870912 \)
- \( x^{90} = (x^3)^{30} = (-2)^{30} = 1073741824 \)
- \( x^{100} = (x^{3})^{33} \cdot x = 1073741824 \cdot x \) (but we will not need the exact value)
- \( x^{103} = (x^{3})^{34} \cdot x = 1073741824 \cdot x \)

### Step 5: Combine the results
Now we can combine the results:
\[ x^{103} + x^{100} + x^{90} + x^{87} + x^{60} + x^{57} + x^{9} + x^{6} + x^{3} + 3 \]
Substituting the values we calculated:
\[ (-2) + 4 + (-8) + 3 + 1048576 + (-2097152) + (-8) + 4 + (-2) + 3 \]
This simplifies to:
\[ -2 + 4 - 8 + 3 + 1048576 - 2097152 - 8 + 4 - 2 + 3 = 2 \]

### Final Result
Thus, the final answer is:
\[ \boxed{2} \]