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Remainder on division by (x - a)(x - b) from f(a) and f(b)

This type of question, asked 2 times: CDS 1 2020, CDS 2 2025. All 2 questions below, oldest first.

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  1. Question 1

    CDS 1 2020

    22
    If f(x)f(x) is divided by (x−α)(x−β)(x - \alpha)(x - \beta) where α≠β\alpha \ne \beta, then what is the remainder ?

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  2. Question 2

    CDS 2 2025

    1
    Let p(x)p(x) be a polynomial. When p(x)p(x) is divided by (x−1)(x-1), it leaves 2 as the remainder. When p(x)p(x) is divided by (x−2)(x-2), it leaves 1 as the remainder. What is the remainder when p(x)p(x) is divided by (x−1)(x−2)(x-1)(x-2)?

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