Question

The remainder, when x²⁰⁰ is divided by x² – 3x + 2 is

(a) (2²⁰⁰ – 1) x + (–2²⁰⁰ + 2) 

(b) (2²⁰⁰ + 1) x + (–2²⁰⁰ – 2) 

(c) (2²⁰⁰ – 1) x + (–2²⁰⁰ – 2) 

(d) 2¹⁰⁰

Answer 1

(a) (2²⁰⁰ – 1) x + (–2²⁰⁰ + 2) 

Let x²⁰⁰ = (x² – 3x + 2). Q(x) + lx + m                 ...(i) 

where, Q(x) = quotient and (lx + m) is the remainder 

Now (x² – 3x + 2) = 0 ⇒ (x – 1) (x – 2) = 0 ⇒ x = 1, 2. 

Substituting x = 1 in (i), we have, 

1²⁰⁰ = 0. Q.(x) + l + m                       ...(ii) 

Similarly, for x = 2, 

2²⁰⁰ = 0. Q(x) + 2l + m                     ...(iii)

∴ l + m = 1, 2l + m = 2²⁰⁰ 

Solving we get, l = 2²⁰⁰ – 1 and m = 2 – 2²⁰⁰ 

Hence remainder = lx + m = (2²⁰⁰ – 1) x + (–2²⁰⁰ + 2).