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Four-term polynomial x^m + x^(m+d) + x^(m+2d) + x^(m+3d) factorised as x^m(1 + x^d)(1 + x^2d)

This type of question, asked 2 times: CDS 1 2024, CDS 2 2024. All 2 questions below, oldest first.

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  1. Question 1

    CDS 1 2024

    33
    Consider the following in respect of the polynomial x4k+x4k+2+x4k+4+x4k+6x^{4k} + x^{4k+2} + x^{4k+4} + x^{4k+6} : 1. The remainder is zero when the polynomial is divided by x2+1x^2 + 1. 2. The remainder is zero when the polynomial is divided by x4+1x^4 + 1. Which of the statements given above is/are correct ?

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  2. Question 2

    CDS 2 2024

    47
    Let kk be a positive integer. What is the quotient when x8k+3+x8k+6+x8k+9+x8k+12x^{8k+3} + x^{8k+6} + x^{8k+9} + x^{8k+12} is divided by (1+x3)(1+x6)(1 + x^3)(1 + x^6) ?

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