Form the polynomial with rational coefficients whose roots are
` i+- sqrt(5)`

Answer

To form the polynomial with rational coefficients whose roots are \( i + \sqrt{5} \) and \( i - \sqrt{5} \), we can follow these steps:

### Step 1: Identify the Roots
The roots given are:
- \( \alpha = i + \sqrt{5} \)
- \( \beta = i - \sqrt{5} \)

### Step 2: Write the Polynomial Equation
The polynomial can be formed using the fact that if \( \alpha \) and \( \beta \) are roots, then the polynomial can be expressed as:
\[ (x - \alpha)(x - \beta) = 0 \]

### Step 3: Substitute the Roots
Substituting the values of \( \alpha \) and \( \beta \):
\[ (x - (i + \sqrt{5}))(x - (i - \sqrt{5})) = 0 \]

### Step 4: Expand the Expression
Now, we expand the expression:
\[ (x - (i + \sqrt{5}))(x - (i - \sqrt{5})) = (x - i - \sqrt{5})(x - i + \sqrt{5}) \]
Using the difference of squares:
\[ = (x - i)^2 - (\sqrt{5})^2 \]

### Step 5: Calculate \( (x - i)^2 \)
Now, calculate \( (x - i)^2 \):
\[ (x - i)^2 = x^2 - 2ix + i^2 \]
Since \( i^2 = -1 \):
\[ = x^2 - 2ix - 1 \]

### Step 6: Substitute Back into the Polynomial
Now substitute back into the polynomial:
\[ = (x^2 - 2ix - 1) - 5 \]
This simplifies to:
\[ = x^2 - 2ix - 6 \]

### Final Polynomial
Thus, the polynomial with rational coefficients is:
\[ x^2 - 2ix - 6 = 0 \]