Single correct option
What is the remainder when x¹⁰⁰ is divided by x² − 3x + 2?
- A. (2¹⁰⁰ − 1)x + (2 − 2¹⁰⁰)
- B. (2¹⁰⁰ − 1)x + (1 − 2¹⁰⁰)
- C. (2¹⁰⁰ + 1)x − 2¹⁰⁰
- D. 2¹⁰⁰x − (2¹⁰⁰ − 1)
Show answer and solution
Answer: (A) (2¹⁰⁰ − 1)x + (2 − 2¹⁰⁰)
The divisor factors as (x − 1)(x − 2) and has degree 2, so the remainder has the form ax + b.
x¹⁰⁰ = (x − 1)(x − 2)Q(x) + ax + b.
Put x = 1: a + b = 1.
Put x = 2: 2a + b = 2¹⁰⁰.
Subtracting: a = 2¹⁰⁰ − 1, and b = 1 − a = 2 − 2¹⁰⁰.
Remainder = (2¹⁰⁰ − 1)x + (2 − 2¹⁰⁰).
Check the other options: at x = 1, option 2 gives 0 instead of 1, and option 4 gives 1 at x = 1 but 2¹⁰⁰ + 1 at x = 2. So they fail.
x¹⁰⁰ = (x − 1)(x − 2)Q(x) + ax + b.
Put x = 1: a + b = 1.
Put x = 2: 2a + b = 2¹⁰⁰.
Subtracting: a = 2¹⁰⁰ − 1, and b = 1 − a = 2 − 2¹⁰⁰.
Remainder = (2¹⁰⁰ − 1)x + (2 − 2¹⁰⁰).
Check the other options: at x = 1, option 2 gives 0 instead of 1, and option 4 gives 1 at x = 1 but 2¹⁰⁰ + 1 at x = 2. So they fail.
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