CDS 2026 (I) Elementary Mathematics, question 7

Topic: Algebra

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Q7

The LCM and HCF of two polynomials p(x)p(x) and q(x)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^{2}x^{2}+a^{4}, then what is q(x)q(x) equal to ?

  1. (a) x4−2a2x2+a4x^{4}-2a^{2}x^{2}+a^{4}
  2. (b) x4+2a2x2+a4x^{4}+2a^{2}x^{2}+a^{4}
  3. (c) x2−a2x^{2}-a^{2}Correct answer
  4. (d) x4−a2x2+a4x^{4}-a^{2}x^{2}+a^{4}

Solution.

Answer: (c) x2−a2x^{2}-a^{2}

  1. Factor p(x)=(x2+ax+a2)(x2−ax+a2)p(x) = \left( x^{2} + ax + a^{2} \right)\left( x^{2} - ax + a^{2} \right) and the stated LCM=(x+a)(x−a)(x2+ax+a2)LCM = (x + a)(x - a)\left( x^{2} + ax + a^{2} \right).
  2. Using p(x)q(x)=HCF×LCMp(x)q(x) = HCF \times LCM gives q(x)=x2−a2q(x) = x^{2} - a^{2}.