The value of expression `x^4-8x^3+18 x^2-8x+2` when `x=2+sqrt(3)`

Options

A. 2
B. 1
C. 0
D. 3

Explanation

To find the value of the expression \( x^4 - 8x^3 + 18x^2 - 8x + 2 \) when \( x = 2 + \sqrt{3} \), we will follow these steps:

### Step 1: Substitute the value of \( x \)
Let \( x = 2 + \sqrt{3} \).

### Step 2: Calculate \( x - 2 \)
We find:
\[ x - 2 = \sqrt{3} \]

### Step 3: Square both sides
Now, we square both sides:
\[ (x - 2)^2 = (\sqrt{3})^2 \]
This gives:
\[ x^2 - 4x + 4 = 3 \]
Rearranging this, we get:
\[ x^2 - 4x + 1 = 0 \quad \text{(1)} \]

### Step 4: Square again
Next, we square \( x - 2 \) again:
\[ (x - 2)^4 = (\sqrt{3})^4 \]
This gives:
\[ (x^2 - 4x + 4)^2 = 9 \]
Expanding the left side:
\[ x^4 - 8x^3 + 16x^2 - 32x + 16 = 9 \]
Rearranging this, we have:
\[ x^4 - 8x^3 + 16x^2 - 32x + 7 = 0 \quad \text{(2)} \]

### Step 5: Substitute back into the original expression
Now, we can substitute back into our original expression:
\[ x^4 - 8x^3 + 18x^2 - 8x + 2 \]
From equation (2), we know:
\[ x^4 - 8x^3 = -16x^2 + 32x - 7 \]
Substituting this into the expression gives:
\[ (-16x^2 + 32x - 7) + 18x^2 - 8x + 2 \]
Combining like terms:
\[ (-16x^2 + 18x^2) + (32x - 8x) + (-7 + 2) = 2x^2 + 24x - 5 \]

### Step 6: Evaluate \( 2x^2 + 24x - 5 \)
Using equation (1) \( x^2 = 4x - 1 \):
\[ 2(4x - 1) + 24x - 5 = 8x - 2 + 24x - 5 = 32x - 7 \]

### Step 7: Substitute \( x = 2 + \sqrt{3} \)
Now substituting \( x = 2 + \sqrt{3} \):
\[ 32(2 + \sqrt{3}) - 7 = 64 + 32\sqrt{3} - 7 = 57 + 32\sqrt{3} \]

### Final Result
Thus, the value of the expression \( x^4 - 8x^3 + 18x^2 - 8x + 2 \) when \( x = 2 + \sqrt{3} \) is:
\[ \boxed{1} \]