Question

Let P(x) be a quadratic polynomial such that p(0) = 1. If p(x) leaves remainder 4 when divided by x - 1 and it leaves remainder 6 when divided by x + 1, then
1. p(-2) = 11
2. p(2) = 11
3. p(2) = 19
4. p(-2) = 19

Answer 1

Correct Answer - Option 4 : p(-2) = 19

Concept:

Remainder theorem: When any function f(x) is divided by (x – a), then the remainder is equal to f(a)

Calculation:

Let us consider  p(x) = ax2 +  bx + c

Substituting x = 0 in this equation, we get

p(0) = 0 + 0 + c = 1

∴ c = 1

When p(x) leaves remainder 4 when divided by x - 1, we get 

x - 1 = 0 

⇒x = 1 

Therefore, p(1) = a + b + c = 4       .....(1) 

Substituting c = 1 in equ (1), we get 

⇒ 4 = a + b + 1 

⇒ a + b = 3                                     .....(2) 

Also, when p(x) leaves remainder 6 when divided by x + 1, we get 

x + 1 = 0 

⇒x = -1 

⇒ p(-1) = a - b + c = 6 

Substituting c = 1 in equ (1), we get 

⇒ a - b = 5                                    .....(3) 

Solving equ (2) and (3), we get 

a = 4, b = -1 

Now  p(x) = 4x2 - x + 1

Put x = -2

p(-2) = 4(-2)2 - (-2) + 1 = 4 × 4 + 2 + 1 

= 16 + 2 + 1 = 19