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

Explanation

Factor the numerator: x^{8k+3}(1 + x^3 + x^6 + x^9). Expanding the denominator gives (1+x^3)(1+x^6) = 1 + x^3 + x^6 + x^9. The polynomials cancel out, leaving the quotient x^{8k+3}.

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