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HCF, LCM and one polynomial given: find the other polynomial

The same question, asked 3 times: CDS 2 2018*, CDS 2 2020*, CDS 1 2026. Every wording below, oldest first.

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  1. Question 1

    CDS 2 2018

    8
    HCF and LCM of two polynomials are (x + 3) and (x3−9x2−x+105)(x^3 - 9x^2 - x + 105) respectively. If one of the two polynomials is x2−4x−21x^2 - 4x - 21, then the other is

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  2. Question 2

    CDS 2 2020

    32
    The HCF and the LCM of two polynomials are 3x+13x + 1 and 30x3+7x2−10x−330x^3 + 7x^2 - 10x - 3 respectively. If one polynomial is 6x2+5x+16x^2 + 5x + 1, then what is the other polynomial ?

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  3. Question 3

    CDS 1 2026

    7
    The LCM and HCF of two polynomials p(x) and q(x) are (x+a)(x3−a3)(x + a)(x^3 - a^3) and (x2−ax+a2)(x^2 - ax + a^2), respectively. If p(x)=x4+a2x2+a4p(x) = x^4 + a^2x^2 + a^4, then what is q(x) equal to ?

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