If `x+(1)/(x)=1` then `x^(52)+x^(46)+x^(32)+x^(26)+x^(21)+x^(15)+x^(6)+x^(3)+4` is equal to

Options

A. 0
B. 3
C. 4
D. 2

Answer

To solve the equation \( x + \frac{1}{x} = 1 \) and find the value of the expression \( x^{52} + x^{46} + x^{32} + x^{26} + x^{21} + x^{15} + x^{6} + x^{3} + 4 \), we can follow these steps:

### Step 1: Rearranging the Initial Equation
Starting with the equation:
\[ x + \frac{1}{x} = 1 \]
We can multiply both sides by \( x \) (assuming \( x \neq 0 \)):
\[ x^2 + 1 = x \]
Rearranging gives us:
\[ x^2 - x + 1 = 0 \]

### Step 2: Finding Roots of the Quadratic Equation
To find the roots of the quadratic equation \( x^2 - x + 1 = 0 \), we can use 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 gives us:
\[ x = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 3: Finding \( x^3 \)
Next, we can find \( x^3 \) using the identity:
\[ x^3 = (x + \frac{1}{x}) \cdot (x^2) - 1 \]
From \( x + \frac{1}{x} = 1 \), we have:
\[ x^3 = 1 \cdot (x^2) - 1 = x^2 - 1 \]
Using \( x^2 = x - 1 \) (from the quadratic equation), we substitute:
\[ x^3 = (x - 1) - 1 = x - 2 \]

### Step 4: Finding Higher Powers of x
We can express higher powers of \( x \) in terms of \( x \) and constants. Notably:
- \( x^3 = -1 \)
- \( x^6 = (x^3)^2 = (-1)^2 = 1 \)

Thus, we can find:
\[ x^{52} = (x^6)^8 \cdot x^4 = 1^8 \cdot x^4 = x^4 \]
\[ x^{46} = (x^6)^7 \cdot x^4 = 1^7 \cdot x^4 = x^4 \]
\[ x^{32} = (x^6)^5 \cdot x^2 = 1^5 \cdot x^2 = x^2 \]
\[ x^{26} = (x^6)^4 \cdot x^2 = 1^4 \cdot x^2 = x^2 \]
\[ x^{21} = (x^6)^3 \cdot x^3 = 1^3 \cdot x^3 = x^3 \]
\[ x^{15} = (x^6)^2 \cdot x^3 = 1^2 \cdot x^3 = x^3 \]
\[ x^{6} = 1 \]
\[ x^{3} = -1 \]

### Step 5: Substituting Back into the Expression
Now substituting these back into the expression:
\[ x^{52} + x^{46} + x^{32} + x^{26} + x^{21} + x^{15} + x^{6} + x^{3} + 4 \]
becomes:
\[ x^4 + x^4 + x^2 + x^2 + x^3 + x^3 + 1 - 1 + 4 \]
Combining like terms:
\[ 2x^4 + 2x^2 + 2(-1) + 4 = 2x^4 + 2x^2 + 2 \]

### Step 6: Finding the Value of \( x^4 \) and \( x^2 \)
Using \( x^2 = x - 1 \):
\[ x^4 = (x^2)^2 = (x - 1)^2 = x^2 - 2x + 1 = (x - 1) - 2x + 1 = -x + 2 \]
Thus:
\[ 2x^4 + 2x^2 + 2 = 2(-x + 2) + 2(x - 1) + 2 = -2x + 4 + 2x - 2 + 2 = 4 \]

### Final Answer
The value of the expression is:
\[ \boxed{4} \]