Form the polynomial equation whose root are
`2 ,1 +- 3i`

Answer

To form the polynomial equation whose roots are \(2\) and \(1 \pm 3i\), we will follow these steps:

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

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

### Step 3: Simplify the complex roots
First, we simplify the product of the complex conjugate roots:
\[ (x - (1 + 3i))(x - (1 - 3i)) \]
This can be rewritten using the difference of squares:
\[ = (x - 1 - 3i)(x - 1 + 3i) = (x - 1)^2 - (3i)^2 \]
Calculating \((3i)^2\):
\[ (3i)^2 = 9i^2 = 9(-1) = -9 \]
Thus, we have:
\[ = (x - 1)^2 + 9 \]

### Step 4: Expand \((x - 1)^2 + 9\)
Now, we expand \((x - 1)^2 + 9\):
\[ (x - 1)^2 = x^2 - 2x + 1 \]
So,
\[ (x - 1)^2 + 9 = x^2 - 2x + 1 + 9 = x^2 - 2x + 10 \]

### Step 5: Combine with the real root
Now we combine this with the real root:
\[ (x - 2)(x^2 - 2x + 10) = 0 \]

### Step 6: Expand the polynomial
Expanding the expression:
\[ x(x^2 - 2x + 10) - 2(x^2 - 2x + 10) = 0 \]
This gives:
\[ x^3 - 2x^2 + 10x - 2x^2 + 4x - 20 = 0 \]
Combining like terms:
\[ x^3 - 4x^2 + 14x - 20 = 0 \]

### Final Polynomial
Thus, the polynomial equation whose roots are \(2\) and \(1 \pm 3i\) is:
\[ x^3 - 4x^2 + 14x - 20 = 0 \]

---