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) = ax2 + 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.