Form the polynomial equation with rational coefficients whose roots are `-sqrt(3)+-isqrt(2)`

Answer

To form the polynomial equation with rational coefficients whose roots are \(-\sqrt{3} + i\sqrt{2}\) and \(-\sqrt{3} - i\sqrt{2}\), we can follow these steps:

### Step 1: Identify the roots
The roots given are:
- \(\alpha = -\sqrt{3} + i\sqrt{2}\)
- \(\beta = -\sqrt{3} - i\sqrt{2}\)

### Step 2: Write the polynomial in factored form
The polynomial can be expressed as:
\[ (x - \alpha)(x - \beta) = 0 \]
Substituting the roots:
\[ (x - (-\sqrt{3} + i\sqrt{2}))(x - (-\sqrt{3} - i\sqrt{2})) = 0 \]
This simplifies to:
\[ (x + \sqrt{3} - i\sqrt{2})(x + \sqrt{3} + i\sqrt{2}) = 0 \]

### Step 3: Expand the polynomial
Now, we will expand the expression:
\[ (x + \sqrt{3} - i\sqrt{2})(x + \sqrt{3} + i\sqrt{2}) \]
This is in the form of \((a - b)(a + b) = a^2 - b^2\), where:
- \(a = x + \sqrt{3}\)
- \(b = i\sqrt{2}\)

Thus, we can write:
\[ = (x + \sqrt{3})^2 - (i\sqrt{2})^2 \]

### Step 4: Calculate \((x + \sqrt{3})^2\)
Calculating \((x + \sqrt{3})^2\):
\[ (x + \sqrt{3})^2 = x^2 + 2\sqrt{3}x + 3 \]

### Step 5: Calculate \((i\sqrt{2})^2\)
Calculating \((i\sqrt{2})^2\):
\[ (i\sqrt{2})^2 = -2 \quad \text{(since } i^2 = -1\text{)} \]

### Step 6: Substitute back into the polynomial
Now substitute back into the polynomial:
\[ (x + \sqrt{3})^2 - (i\sqrt{2})^2 = x^2 + 2\sqrt{3}x + 3 - (-2) \]
This simplifies to:
\[ x^2 + 2\sqrt{3}x + 3 + 2 = x^2 + 2\sqrt{3}x + 5 \]

### Step 7: Final polynomial equation
Thus, the polynomial equation with rational coefficients is:
\[ x^2 + 2\sqrt{3}x + 5 = 0 \]