Question

What is the remainder when the polynomial p(x) = x²⁰⁰ – 2x¹⁹⁹ + x⁵⁰ – 2x⁴⁹ + x² + x + 1 is divided by (x – 1) (x – 2) ? 

(1) 1 

(2) 7 

(3) 2x + 1 

(4) 6x – 5 

Answer 1

Correct option (4) 6x – 5

Explanation:

f(x) = x²⁰⁰ – 2x¹⁹⁹ + x⁵⁰ –2x⁴⁹ + x² + x + 1 

f(x) = (x – 1)(x – 2) q(x) + ax + b

1 – 2 + 1 – 2 + 1 + 1 + 1 = a + b

1 = a + b

2²⁰⁰ – 2²⁰⁰ + 2⁵ – 2⁵⁰ + 4 + 2 + 1 = 2a + b 

7 = 2a + b

– 6 = – a ⇒ a = 6

∴ b = –5

∴ 6b = –5