What is the remainder when the polynomial x¹⁰⁰ – 2x⁵¹ + 1 is divided by (x² – 1) ?

Correct option is C

​$$\textbf{Given:} \\f(x) = x^{100} - 2x^{51} + 1 \\\text{Divisor} = x^2 - 1 \\\textbf{Concept Used:} \\\text{Remainder theorem for quadratic divisor} \\\textbf{Formula Used:} \\\text{If } f(x) \text{ is divided by } (x^2-1), \\\text{then remainder } = ax + b \\\text{where } x^2-1 = 0 \Rightarrow x = 1,-1 \\\textbf{Solution:} \\\text{Let remainder } = ax + b \\f(1) = a(1) + b \\f(-1) = a(-1) + b \\\text{Compute } f(1): \\f(1) = 1^{100} - 2(1)^{51} + 1 \\= 1 - 2 + 1 \\= 0 \\\Rightarrow a + b = 0 \quad (1) \\\text{Compute } f(-1): \\f(-1) = (-1)^{100} - 2(-1)^{51} + 1 \\= 1 - 2(-1) + 1 \\= 1 + 2 + 1 \\= 4 \\\Rightarrow -a + b = 4 \quad (2) \\\text{Solve (1) and (2):} \\a + b = 0 \\-a + b = 4 \\\text{Subtract (1) from (2):} \\-2a = 4 \\a = -2 \\b = 2 \\\text{Remainder} = ax + b \\= -2x + 2 \\\textbf{Final Answer:} \\-2x + 2$$​​