Form the polynomial equation whose root are
`1,3-sqrt(-2)`

Answer

To form the polynomial equation whose roots are \(1\) and \(3 - \sqrt{-2}\), we will follow these steps:

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

### Step 2: Write the polynomial in factored form
The polynomial can be expressed in factored form as:
\[ (x - \alpha)(x - \beta) = 0 \]
Substituting the values of \(\alpha\) and \(\beta\):
\[ (x - 1)(x - (3 - i\sqrt{2})) = 0 \]

### Step 3: Expand the polynomial
Now, we will expand the expression:
\[ (x - 1)(x - (3 - i\sqrt{2})) = (x - 1)(x - 3 + i\sqrt{2}) \]

Using the distributive property (FOIL method):
\[ = x(x - 3 + i\sqrt{2}) - 1(x - 3 + i\sqrt{2}) \]
\[ = x^2 - 3x + i\sqrt{2}x - x + 3 - i\sqrt{2} \]
\[ = x^2 - 4x + 3 + i\sqrt{2}x - i\sqrt{2} \]

### Step 4: Combine like terms
Now, we combine the real and imaginary parts:
\[ = x^2 - 4x + (3 - i\sqrt{2}) + i\sqrt{2}x \]

Since we want a polynomial with real coefficients, we need to consider the conjugate of the root \(3 - i\sqrt{2}\), which is \(3 + i\sqrt{2}\). Thus, we will also include this root.

### Step 5: Form the complete polynomial
Now we will form the polynomial with both roots \(1\) and \(3 \pm i\sqrt{2}\):
\[ (x - 1)((x - 3) + i\sqrt{2})((x - 3) - i\sqrt{2}) = 0 \]

The product of the conjugate pair can be simplified:
\[ (x - 3)^2 + 2 = 0 \]
Thus, the polynomial becomes:
\[ (x - 1)((x - 3)^2 + 2) = 0 \]

### Step 6: Expand the final polynomial
Now we expand this:
\[ = (x - 1)(x^2 - 6x + 9 + 2) \]
\[ = (x - 1)(x^2 - 6x + 11) \]
Now we expand:
\[ = x(x^2 - 6x + 11) - 1(x^2 - 6x + 11) \]
\[ = x^3 - 6x^2 + 11x - x^2 + 6x - 11 \]
\[ = x^3 - 7x^2 + 17x - 11 \]

### Final Polynomial
Thus, the polynomial equation whose roots are \(1\) and \(3 - \sqrt{-2}\) is:
\[ x^3 - 7x^2 + 17x - 11 = 0 \]