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1 - x - x^n + x^(n+1) is divisible by (1 - x)^2

The same question, asked 2 times: CDS 2 2020, CDS 1 2023. Every wording below, oldest first.

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  1. Question 1

    CDS 2 2020

    39
    1−x−xn+xn+11 - x - x^n + x^{n+1}, where n is a natural number, is divisible by

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  2. Question 2

    CDS 1 2023

    43
    Consider the following statements in respect of the polynomial 1−x−xn+xn+11 - x - x^n + x^{n+1}, where nn is a natural number : 1. It is divisible by 1−2x+x21 - 2x + x^2. 2. It is divisible by 1−xn1 - x^n. Which of the statements given above is/are correct ?

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