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Add or subtract several algebraic fractions by factorising the denominators

This type of question, asked 4 times: CDS 2 2020, CDS 1 2021, CDS 2 2023, CDS 2 2025. All 4 questions below, oldest first.

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  1. Question 1

    CDS 2 2020

    34
    What is the value of a2+aca2c−c3−a2−c2a2c+2ac2+c3−2ca2−c2+3a+c\dfrac{a^2 + ac}{a^2c - c^3} - \dfrac{a^2 - c^2}{a^2c + 2ac^2 + c^3} - \dfrac{2c}{a^2 - c^2} + \dfrac{3}{a + c} ?

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  2. Question 2

    CDS 1 2021

    17
    What is 8x1−x4−4xx2+1+x+1x−1−x−1x+1\frac{8x}{1 - x^4} - \frac{4x}{x^2 + 1} + \frac{x + 1}{x - 1} - \frac{x - 1}{x + 1} equal to?

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  3. Question 3

    CDS 2 2023

    69
    What is x2−y2−z2−2yzx2+y2−z2+2xy+x2−y2−z2−2yzx2−y2+z2−2xz\frac{x^2 - y^2 - z^2 - 2yz}{x^2 + y^2 - z^2 + 2xy} + \frac{x^2 - y^2 - z^2 - 2yz}{x^2 - y^2 + z^2 - 2xz} equal to ?

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  4. Question 4

    CDS 2 2025

    21
    What is (a+b)2(c−a)(c+a+b)+(a+b)cc2+bc−a2−ab−(a+2b+c)2(c−a)\frac{(a+b)^2}{(c-a)(c+a+b)}+\frac{(a+b)c}{c^2+bc-a^2-ab}-\frac{(a+2b+c)}{2(c-a)}, a≠ba\ne b, b≠cb\ne c, c≠ac\ne a equal to?

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