Correct Answer - Option 2 : -2x

^{2} + 6x

__Given:__

Polynomial is quadratic, and

Remainder = 4, when divided by (x - 1),

Remainder = 4, when divided by (x - 2)

Remainder = 0, when divided by (x - 3)

**Concept:**

**Remainder theorem:**

If a **polynomial P(x)** is **divided by (x - a)**, then the **remainder** of the polynomial will be** P(a)**.

**Calculation**

Let quadratic polynomail is

P(x) = ax^{2} + bx + c

Given that when P(x) is divided by (x - 1), the remainder comes 4, so by using the reminder theorem

As, a = 1

⇒ P(1) = 4

⇒ a(1)^{2 }+ b(1) + c = 4

⇒ a + b + c = 4

⇒ c = 4 - a - b ---(1)

Simmilarly, for (x - 2), a = 2

⇒ P(2) = 4

⇒ a(2)2 + b(2) + c = 4

⇒ 4a + 2b + c = 4

From equation first,

4a + 2b + 4 - a - b = 4

⇒ 3a + b = 0

⇒ b = - 3a ---(2)

Simmilarly, for (x - 3), a = 3

⇒ a(3)2 + b(3) + c = 0

⇒ 9a + 3b + c = 0

⇒ 8a + 2b = -4

⇒ 4a + b = -2

⇒ 4a - 3a = -2 From equation (2)

⇒ a = -2

⇒ b = 6 From equation (2)

⇒ c = 4 - (-2) - 6 = 0 From equation (1)

Hence, **required polynomial **will be** -2x2 + 6x.**