The biquadratic equation, two of whose roots are 1 + `i` , 1 - `sqrt(2)` is

Options

A. `x^4 -4x^3 +5x^2 -2x -2=0`
B. `x^4 -4x^3 -5x^2 +2x +2=0`
C. `x^4 +4x^3 -5x^2 +2x -2=0`
D. `x^4 +4x^3 -5x^2 -2x+2=0`

Answer

To find the biquadratic equation given two of its roots, we can follow these steps:

### Step 1: Identify the given roots
The roots provided are:
- \( r_1 = 1 + i \)
- \( r_2 = 1 - \sqrt{2} \)

### Step 2: Determine the other roots
Since complex roots and irrational roots come in conjugate pairs, the other two roots will be:
- \( r_3 = 1 - i \) (conjugate of \( r_1 \))
- \( r_4 = 1 + \sqrt{2} \) (conjugate of \( r_2 \))

### Step 3: Formulate the factors of the biquadratic equation
The biquadratic equation can be expressed as the product of two quadratic equations formed by these roots:
\[ (x - r_1)(x - r_2)(x - r_3)(x - r_4) = 0 \]

### Step 4: Write the quadratic factors
The factors can be grouped as follows:
1. For the roots \( r_1 \) and \( r_3 \):
\[ (x - (1 + i))(x - (1 - i)) \]
This simplifies to:
\[ (x - 1 - i)(x - 1 + i) = (x - 1)^2 - i^2 = (x - 1)^2 + 1 = x^2 - 2x + 2 \]

2. For the roots \( r_2 \) and \( r_4 \):
\[ (x - (1 - \sqrt{2}))(x - (1 + \sqrt{2})) \]
This simplifies to:
\[ (x - 1 + \sqrt{2})(x - 1 - \sqrt{2}) = (x - 1)^2 - (\sqrt{2})^2 = (x - 1)^2 - 2 = x^2 - 2x - 1 \]

### Step 5: Combine the quadratic factors
Now, we combine the two quadratic equations:
\[ (x^2 - 2x + 2)(x^2 - 2x - 1) = 0 \]

### Step 6: Expand the product
To find the biquadratic equation, we expand the product:
\[ (x^2 - 2x + 2)(x^2 - 2x - 1) \]
Using the distributive property:
\[ = x^4 - 2x^3 - x^2 - 2x^3 + 4x^2 + 2x + 2x^2 - 4x - 2 \]
Combining like terms:
\[ = x^4 - 4x^3 + 5x^2 - 2x - 2 \]

### Final Biquadratic Equation
Thus, the biquadratic equation is:
\[ x^4 - 4x^3 + 5x^2 - 2x - 2 = 0 \]