Remainder and Factor Theorem

Answer : A

Since f(x) is divisible by x - 1, f(x) = (x - 1)Q(x) where Q(x) is the quotient.
Then, f(x - 1) = (x - 1 - 1)Q(x)
We get f(x - 1) = (x - 2)Q(x)
Hence, f(x - 1) is divisible by x - 2.

Answer : B

Let f(x) =  x³ - x² + 1
First of all, we solve 2x + 1 = 0. We get x = -1/2,
By remainder theorem, the remainder should be f(-1/2) = 5/8

Challenging question:

Let y = f(x). When f(x) is divided by x-3, the remainder is 2. What is the remainder when [f(x)]⁵  is divided by x - 3?

Answer : 32

Let g(x) = [f(x)]⁵
The required remainder is g(3) = [f(3)]⁵ = 2⁵ = 32