## Finding the Polynomial from its Roots

Given the roots of a polynomial, we can construct the polynomial itself. This is based on the **Factor Theorem**, which states that a polynomial has a factor (x - a) if and only if 'a' is a root of the polynomial.

We are given the roots:

**x = 3****x = i****x = 2 + i**

From these roots, we can construct the factors:

**(x - 3)****(x - i)****(x - (2 + i))**

Multiplying these factors together will give us the polynomial:

**(x - 3)(x - i)(x - (2 + i))**

To find the polynomial, we need to expand this expression:

**Step 1:** Expand the first two factors.

(x - 3)(x - i) = x² - ix - 3x + 3i = **x² - (3 + i)x + 3i**

**Step 2:** Multiply the result from Step 1 with the third factor.

(x² - (3 + i)x + 3i)(x - (2 + i)) = x³ - (2 + i)x² - (3 + i)x² + (3 + i)(2 + i)x + 3ix - 3i(2 + i)

**Step 3:** Simplify the expression.

x³ - (2 + i + 3 + i)x² + (6 + 3i + 2i - i²)x + 3ix - 6i - 3i²

**Step 4:** Remember that i² = -1.

x³ - (5 + 2i)x² + (8 + 5i)x + 3ix + 6 - 3

**Step 5:** Combine like terms.

**x³ - (5 + 2i)x² + (8 + 8i)x + 6**

Therefore, the polynomial with the given roots is **x³ - (5 + 2i)x² + (8 + 8i)x + 6**.

**Note:** This polynomial has complex coefficients due to the complex roots.