CDS 2026 (I) Elementary Mathematics, question 82

Topic: Algebra

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Passage · Q81 to Q82

x4+px3+qx2+x+6x^{4}+px^{3}+qx^{2}+x+6 is divisible by x2−x−6x^{2}-x-6.

Q82

What is the value of qq ?

  1. (a) −7-7Correct answer
  2. (b) −5-5
  3. (c) −3-3
  4. (d) −1-1

Solution.

Answer: (a) −7-7

  1. Using the same roots of the divisor, F(3)=0F(3) = 0 and F(−2)=0F( - 2) = 0 give 3p+q=−103p + q = - 10 and −2p+q=−5- 2p + q = - 5.
  2. Hence p=−1p = - 1 and q=−7q = - 7.
  3. Check: x4−x3−7x2+x+6=(x2−x−6)(x2−1)x^{4} - x^{3} - 7x^{2} + x + 6 = \left( x^{2} - x - 6 \right)\left( x^{2} - 1 \right).