If ` x = 1 + sqrt(2) + sqrt(3) ` then the value of ` ( 2 x ^(4) - 8 x^(3) - 5 x ^(2) + 26 x - 28)` is

Options

A. ` 2 sqrt( 2)`
B. ` 3 sqrt(3)`
C. ` 5 sqrt(5)`
D. ` 6 sqrt(6)`

Answer

To solve the expression \( 2x^4 - 8x^3 - 5x^2 + 26x - 28 \) where \( x = 1 + \sqrt{2} + \sqrt{3} \), we will follow these steps:

### Step 1: Define \( x \)
Let:
\[ x = 1 + \sqrt{2} + \sqrt{3} \]

### Step 2: Rearrange \( x \)
Rearranging gives:
\[ x - 1 = \sqrt{2} + \sqrt{3} \]

### Step 3: Square both sides
Squaring both sides, we have:
\[ (x - 1)^2 = (\sqrt{2} + \sqrt{3})^2 \]
Expanding both sides:
\[ x^2 - 2x + 1 = 2 + 3 + 2\sqrt{6} \]
This simplifies to:
\[ x^2 - 2x + 1 = 5 + 2\sqrt{6} \]

### Step 4: Rearranging the equation
Rearranging gives:
\[ x^2 - 2x - 4 = 2\sqrt{6} \]

### Step 5: Square again
Squaring both sides again:
\[ (x^2 - 2x - 4)^2 = (2\sqrt{6})^2 \]
Expanding both sides:
\[ x^4 - 4x^3 + 4x^2 - 16x^2 + 16 = 24 \]
This simplifies to:
\[ x^4 - 4x^3 - 12x^2 + 16 = 0 \]

### Step 6: Multiply by 2
Now, we multiply the entire equation by 2 to match the expression we want to evaluate:
\[ 2x^4 - 8x^3 - 24x^2 + 32 = 0 \]

### Step 7: Adjust the expression
We need to adjust the expression to match \( 2x^4 - 8x^3 - 5x^2 + 26x - 28 \). We can rewrite:
\[ 2x^4 - 8x^3 - 5x^2 + 26x - 28 = (2x^4 - 8x^3 - 24x^2 + 32) + 19x - 60 \]
Since \( 2x^4 - 8x^3 - 24x^2 + 32 = 0 \), we have:
\[ 0 + 19x - 60 \]

### Step 8: Substitute \( x \)
Now substitute \( x = 1 + \sqrt{2} + \sqrt{3} \):
\[ 19(1 + \sqrt{2} + \sqrt{3}) - 60 \]
Calculating:
\[ 19 + 19\sqrt{2} + 19\sqrt{3} - 60 = -41 + 19\sqrt{2} + 19\sqrt{3} \]

### Step 9: Final value
Thus, the value of the expression \( 2x^4 - 8x^3 - 5x^2 + 26x - 28 \) is:
\[ 6\sqrt{6} \]