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Two roots (or a quadratic factor) given: find two unknown coefficients

This type of question, asked 4 times: CDS 1 2020, CDS 1 2025, CDS 1 2026, CDS 2 2026. All 5 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    CDS 1 2020

    49
    If (x2−1)(x^2 - 1) is a factor of ax4+bx3+cx2+dx+eax^4 + bx^3 + cx^2 + dx + e, then which one of the following is correct?

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  2. Question 2

    CDS 1 2020

    13
    What are the values of pp and qq respectively, if (x−1)(x - 1) and (x+2)(x + 2) divide the polynomial x3+4x2+px+qx^{3} + 4x^{2} + px + q ?

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  3. Question 3

    CDS 1 2025

    81
    If x2−5x+4x^2 - 5x + 4 is a factor of x4−px2+qx^4 - px^2 + q, then what are the values of p and q respectively ?

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  4. Question 4 (case study, 2 parts)

    CDS 1 2026

    For the next two (02) items that follow : x4+px3+qx2+x+6x^4 + px^3 + qx^2 + x + 6 is divisible by x2−x−6x^2 - x - 6.
    81
    What is the value of p ?

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    82
    What is the value of q ?

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  5. Question 5

    CDS 2 2026

    3
    If x2−4x+3x^2 - 4x + 3 is a factor of x4+px2+qx^4 + px^2 + q, then what is (p−q)(p - q) equal to ?

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