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Polynomial factor problem

CMAT 2021 · Quantitative Techniques & Data Interpretation · Medium · Algebra

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

  1. A.

    (5,2)

  2. B.

    (5,4)

  3. C.

    (-5,-4)

  4. D.

    (-5,4)

Answer

B

Explanation

x2−3x+2=(x−1)(x−2)x^2 - 3x + 2 = (x-1)(x-2).

Since this is a factor of f(x)=x4−px2+qf(x) = x^4 - px^2 + q, f(1)=0f(1) = 0 and f(2)=0f(2) = 0.

  1. f(1)=1−p+q=0  ⟹  p−q=1f(1) = 1 - p + q = 0 \implies p - q = 1
  2. f(2)=16−4p+q=0  ⟹  4p−q=16f(2) = 16 - 4p + q = 0 \implies 4p - q = 16

Subtracting (1) from (2): 3p=15  ⟹  p=53p = 15 \implies p = 5 q=5−1=4q = 5 - 1 = 4

Thus, (p,q)=(5,4)(p,q) = (5,4).

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