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What should be added to one rational expression to get another

This type of question, asked 2 times: CDS 1 2021, CDS 2 2026. All 2 questions below, oldest first.

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  1. Question 1

    CDS 1 2021

    12
    What should be added to 1(x−2)(x−4)\frac{1}{(x - 2)(x - 4)} to get 2x−5(x2−5x+6)(x−4)\frac{2x - 5}{(x^2 - 5x + 6)(x - 4)}?

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  2. Question 2

    CDS 2 2026

    34
    What should be added to x2−1x2+1\frac{x^2 - 1}{x^2 + 1} to get 2x3+3x2−1(x+2)(x2+1)\frac{2x^3 + 3x^2 - 1}{(x + 2)(x^2 + 1)} ?

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