Solve the equation
` x^4 +2x^3 -5x^2 +6x +2=0` given that `1+i` is a root

Answer

To solve the equation \( x^4 + 2x^3 - 5x^2 + 6x + 2 = 0 \) given that \( 1 + i \) is a root, we can follow these steps:

### Step 1: Identify the conjugate root
Since \( 1 + i \) is a root and the coefficients of the polynomial are real, its conjugate \( 1 - i \) must also be a root.

### Step 2: Form a quadratic factor
The roots \( 1 + i \) and \( 1 - i \) can be used to form a quadratic factor:
\[ (x - (1 + i))(x - (1 - i)) = (x - 1 - i)(x - 1 + i) \]
Using the difference of squares:
\[ = (x - 1)^2 - (i)^2 = (x - 1)^2 + 1 \]
Expanding this:
\[ = (x^2 - 2x + 1 + 1) = x^2 - 2x + 2 \]

### Step 3: Divide the original polynomial by the quadratic factor
Now, we will divide the original polynomial \( f(x) = x^4 + 2x^3 - 5x^2 + 6x + 2 \) by \( x^2 - 2x + 2 \).

Using polynomial long division:
1. Divide the leading term: \( x^4 \div x^2 = x^2 \)
2. Multiply \( x^2 \) by \( x^2 - 2x + 2 \) and subtract from \( f(x) \):
\[ x^4 + 2x^3 - 5x^2 + 6x + 2 - (x^4 - 2x^3 + 2x^2) = 4x^3 - 7x^2 + 6x + 2 \]
3. Repeat the process:
- Divide \( 4x^3 \div x^2 = 4x \)
- Multiply \( 4x \) by \( x^2 - 2x + 2 \) and subtract:
\[ 4x^3 - 8x^2 + 8x \]
\[ 4x^3 - 7x^2 + 6x + 2 - (4x^3 - 8x^2 + 8x) = x^2 - 2x + 2 \]
4. Finally, divide \( x^2 - 2x + 2 \) by \( x^2 - 2x + 2 \) which gives a remainder of 0.

Thus, we have:
\[ f(x) = (x^2 - 2x + 2)(x^2 + 4x + 1) \]

### Step 4: Solve the remaining quadratic equation
Now we need to solve \( x^2 + 4x + 1 = 0 \) using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 4, c = 1 \):
\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm \sqrt{12}}{2} \]
\[ = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3} \]

### Step 5: Compile all roots
The roots of the equation \( x^4 + 2x^3 - 5x^2 + 6x + 2 = 0 \) are:
1. \( 1 + i \)
2. \( 1 - i \)
3. \( -2 + \sqrt{3} \)
4. \( -2 - \sqrt{3} \)

### Final Answer
The roots of the equation are:
\[ 1 + i, \quad 1 - i, \quad -2 + \sqrt{3}, \quad -2 - \sqrt{3} \]